In addition to the older functionality for formal desingularisation of hypersurfaces in P3 in characteristic zero described in the next section, there is now (Magma V2.21) a more general desingularisation routine available that desingularises by blowing up. It has the current restriction that it only applies to surfaces with point singularities (the singular subscheme is of dimension zero) but this should only be a provisional restriction that is lifted in later releases.
The advantages are firstly that it applies to surfaces in any ambient of any characteristic with the mild proviso that all of the singular points have to lie in definable affine patches. In particular, it applies to surfaces in affine, projective or product projective space of any dimension. Secondly, it provides a much more detailed description than formal desingularisation. Global information is available for each irreducible divisor lying over each singular point rather than just generic information. This allows the computation of the intersection pairing between these blow-up divisors as well as computation of intersection pairings between the blow-up divisors and strict transforms (pull-backs to the desingularisation with blow-up divisor components removed) of divisors on the singular surface model. This is in addition to the functionality for computing the multiplicity of blow-up divisors in the pull-backs of divisors. The latter should be available for formal desingularisation also but is not fully implemented currently. There is much useful information regarding the canonical divisor on the desingularisation that has been extended from hypersurfaces only to any ordinary projective surface in Magma V2.25. Finally, it has the advantage that it is often much faster than formal desingularisation in the ordinary hypersurface case where both apply. In the main resolution intrinsic ResolveSingularSurface, it is now used as the default method, even in the hypersurface case, as long as the surface has only point singularities. We will illustrate the speed and rich range of functionality in three extended examples at the end of this section.
Let k be the base field of the surface S. In resolving by blow-up, there is a choice to make. When blowing up a zero-dimensional reduced subscheme, we can choose to blow up the whole thing in one operation; split into irreducible components, which represent a conjugate set of points in S(bar(k)), and blow these up separately; or keep base-changing to fields over which all points are rational and blow up only rational points. We have chosen the second method. Splitting into irreducible components has some overhead cost, but we cannot identify the different irreducible blow-up components at the end without it. We also do not base-change in order to blow up only rational points, because it is probably best to stick to working over the original ground field k. Groebner basis computations in Magma are very highly-tuned over the rational numbers, for example, but may not work quite so efficiently over number fields. On the other hand, blowing up an irreducible zero-dimensional subscheme rather than the points over an extension field can lead to an increase in the number of variables for the ambients that our blow-up patches lie in. However, we usually have to blow up singular curves at some stage in the resolution process and it is hard to avoid a potential increase in the number of variables in these operations, anyway. In addition, when we avoid field extensions, the final `irreducible' blow-up components over a singular point will often not all be geometrically irreducible (never in the case of blowing up a `non-rational singular point'), but some will split into a set of conjugate irreducible components over bar(k). This, however, should not cause many problems. For example, multiplicities of conjugate blow-up divisors in rational divisors or differentials are all the same, so they always enter into decompositions of rational divisors as sums of our blow-up components with multiplicity. If the user really needs to inspect the blow-up divisors in detail, rather than just using the top-level intrinsics, there are functions to return affine patches of them and they can be used to check for geometric irreducibility or not. All of the package intrinsics only require irreducible over k blow-up components for their implementation.
AdjComp: BoolElt Default: false
UseFormalDesing: BoolElt Default: false
SetVerbose("Resolve", n): Maximum: 1
The main intrinsic to compute a desingularisation of surface S and return a data list containing information about it. The desingularisation is either formal or of blow-up type. The second return value is a code value for the type: 1 for formal and 2 for blow-up desingularisation. The type of desingularisation is determined as follows. If the characteristic of the base field k is not 0, or S does not lie in ordinary projective space of dimension 3, P3, only blow-up desingularisation is possible. However, this currently requires S to have only point singularities (its singular subscheme must be empty or of dimension 0). If only blow-up type is possible and S has curves in its singular locus, a runtime error will result. When the ambient of S is P3, k is of characteristic 0 and the singularities are of dimension 0, either method could be used but blow-up is taken as the default. To force formal desingularisation in the case of a choice, the parameter UseFormalDesing can be set to true (the default is false). If the user only wishes to perform adjoint computations (see HomAdjoints), determine the arithmetic, geometric or pluri-genera of the desingularisation, or perform most other computations that only depend on the canonical divisor class of the desingularisation, some time may be saved by setting parameter AdjComp to true. This means that the resolution is not fully carried out, but the only singularities in the desingularisation process that are not fully resolved are of `canonical type': none of the resolution curves above them occur in the support of some canonical divisor and they do not impose conditions on the sections of the canonical sheaf. For blow-up resolution, it is only simple singularities on the base singular scheme S that are ignored when AdjComp is true (we do not check for ignorable singularities arising at later stages) and it probably saves little time using this option in general, as simple singularities should resolve quite quickly under blow-up. However, it can save more time with formal desingularisation.In neither case is the resolution guaranteed to be a minimal one. That is, there may exist (-1)-curves in the irreducible components lying above singularities.
The elements of the list returned are of type DesingData for blow-up resolution and tuples containing a sequence of four algebraic power series and a multiplicity for formal desingularisation. In the formal case, there is a tuple for each exceptional divisor (lying over a singular point or curve on S) of the desingularisation and the power series define the map from the completion of the local ring of the generic point of that divisor to S. This is all described more fully in the next section on surfaces in P3. For blow-up desingularisation, there is a DesingData object for each irreducible component p of the singular subscheme of S (these correspond to a conjugate set of singular points in S(bar(k)) - we will usually just refer to each one as a singular point from now on, whether it is a k-rational singular point or a set of conjugates). Each object contains a set of affine patches arising in the blow-up process along with birational blow-down maps to a chosen affine patch of S containing p; a set of irreducible curves for each patch which give the intersection of the blow-up divisors with that patch; and information to determine which irreducible curves for different patches are parts of the same complete irreducible blow-up divisor over p. They also cache other useful information about the resolution over p that has been determined by post-resolution intrinsic calls.
Once a formal or blow-up resolution has been computed, it is stored in an attribute of S for use in other intrinsic calls or to be returned by subsequent calls to this intrinsic. Resolution data for both formal and blow-up decomposition will be stored if resolutions of both types are constructed at different times. If a blow-up data list is computed and stored for Adjoint computation only and a subsequent call to ResolveSingularSurface or equivalent is made with AdjComp false, the stored resolution data is upgraded to full resolution data by computing blow-up resolutions over the simple singularities only: the entire resolution process is not repeated.
AdjComp: BoolElt Default: false
SetVerbose("Resolve", n): Maximum: 1
This is just a variation of intrinsic ResolveSingularSurface above that always uses blow-up desingularisation. The conditions on S are the same and the returned resolution list is again cached.
SetVerbose("Resolve", n): Maximum: 1
This is another variant that we have exported. In this case, X must be an affine surface and Y an irreducible, reduced subscheme of dimension zero, which is a single singular `point' (i.e. a conjugate set of points over bar(k)) on X. Note that the conditions on Y are not checked here. It performs the blow-up resolution of just Y and returns the desingularisation data object for it. We have exported this so the user can have a look at the desingularisation of individual singularities in case the overall desingularisation process is a little slow. If the original surface is a projective one, it should be applied to an affine patch that contains the singularity.
Functions to obtain some basic information about the blow-up resolution over a singular point. These are not needed for the main magma functionality related to desingularisation but they are provided for times when the user may wish to examine the local structure of the resolution in more detail.
The number of irreducible divisors (curves) lying over the surface singular point p for which dsd is the desingularisation data object.
The singular point on a surface S for which dsd is the blow-up desingularisation object. Note that the `point' is not necessarily a point of S(k), where k is the base field of S. It is a reduced, irreducible (zero-dimensional) component of the singular locus of S and so is a scheme-theoretic point representing a conjugate set of singular points of S(bar(k)). For this reason it is returned as a subscheme of S rather than as a point type.
Given dsd, the desingularisation data object for the blow-up resolution of point singularity p on surface S, returns an affine curve C giving one patch of the i-th irreducible divisor lying over p, the surface blow-up affine patch Sa in which it lies, and the blow-down map of Sa to S.
Let dsd be the desingularisation data object for the blow-up resolution of point singularity p on surface S. This intrinsic returns the most detailed information about the i-th irreducible divisor lying over p. It gives a full set of affine patches in the blow-up resolution of p that cover the complete divisor. As it involves a rather complicated set-up, this intrinsic is best avoided unless the user really knows what they are doing. The description and comments below should be read with care.The first actual return value is a list of 4-tuples, one for each patch. The entries of the tuples are
The patch index of the affine patch of S (containing p), which is the Codomain of all of the maps which are third entries of a tuple, is the second return value. If S is itself affine, this second return value is just 0.
- (i)
- An affine curve C giving the irreducible divisor on the patch.
- (ii)
- The affine surface Sa which is the blow-up patch in which C lies.
- (iii)
- The birational blow-down map of Sa to an affine patch of S in which p lies.
- (iv)
- The index of Sa in the full list of blow-up patches in dsd.
We choose to gives maps to the base patch here, rather than to S as in the last intrinsic, because it makes it slightly more efficient to compose one with the inverse of another to get the rational map from one curve patch to another. However, it should be noted that the rational maps between patches given directly as the blow-down composed with the blow-up do not generally restrict to maximally-defined maps on the affine curves and may have the entire affine curve in their base-scheme in some cases.
Also, we note that not all blow-up patches of the surface are non-singular. So as not to do redundant work and to avoid multiple copies of blow-up divisors, we only blow up one copy of isolated singular points, which will generally occur in multiple blow-up affine patches (these affine patches all intersect abstractly in Zariski open subsets) that arise in the resolution process. However, over the (abstractly intersecting) affine surface patches containing the complete irreducible blow-up divisor, each point on the divisor will occur on at least one patch at a point where the surface is non-singular. It is on these patches (relative to the particular point) that the local description of the divisor at the point is the correct one.
The package intrinsics dealing with blow-up divisors are coded to get around these problems, so it is recommended that these be used for required operations if possible.
> P5<a,b,c,d,e,f> := ProjectiveSpace(Rationals(),5); > S := Surface(P5,[b^2-a*d, b*c-a*f, c^2-b*e, c*d-b*f, d*e-c*f]); > HasOnlySimpleSingularities(S : ReturnList := true); true [* <(0 : 0 : 0 : 0 : 1 : 0), "A", 2>, <(1 : 0 : 0 : 0 : 0 : 0), "A", 1> *] > dsds := ResolveSingularSurface(S); > #dsds; //one for each singular point 2 > [NumberOfBlowUpDivisors(dsd) : dsd in dsds]; [2,1] > // First dsd corresponds to the A2. > dsd := dsds[1]; > SingularPoint(dsd); Scheme over Rational Field defined by a, b, c, d, f > Da,Sa,prj := BlowUpDivisor(S, dsd, 1); > Da; Ambient(Da); Curve over Rational Field defined by $.1 Affine Space of dimension 2 over Rational Field Variables: $.1, $.2 > Sa; // blow-up surface patch is just A^2! Scheme over Rational Field defined by 0 > prj; //typical blow-up map! Mapping from: Sch: Sa to Srfc: X5 with equations : $.1^2*$.2 $.1^2*$.2^2 $.1*$.2 $.1^2*$.2^3 1 $.1*$.2^2 > // get patch cover of the full blow-up divisor D, which is just P^1 > lst := BlowUpDivisorAllPatches(dsd, 1); > #lst; //2 patch cover 2 > D1 := lst[1][1]; D2 := lst[2][1]; > mp12 := Restriction(Expand(lst[1][3] * Inverse(lst[2][3])),D1,D2); > A1<x,y> := Ambient(D1); A2<x,y> := Ambient(D2); > // D1 and D2 are both the line x=0 in an affine plane. The rational map > // between them equates to the usual y -> 1/y > D1; D2; mp12; Curve over Rational Field defined by x Curve over Rational Field defined by x Mapping from: CrvPln: D1 to CrvPln: D2 with equations : $.1*$.2^2 1/$.2
This subsection deals with blow-up desingularisation intrinsics to do with multiplicities of irreducible blow-up divisors in the pullbacks of functions and divisors, and intersection numbers for divisors that are blow-up divisors or the strict transforms of singular surface divisors on the desingularised surface. Those involving only blow-up divisors depend only on the desingularisation object and work in any ambient. Those that involve divisors on the singular surface work for surfaces in ordinary projective space. All intrinsics use a DesingData desingularisation data object of blow-up type cached in singular surface S. If the resolution process hasn't already been carried out, it is performed at the start (and the result cached).
get_self_ints: BoolElt Default: true
SetVerbose("MultsAndInts", n): Maximum: 1
Returns the intersection matrix for the irreducible (blow-up) divisors that lie above the surface singular point p whose resolution is represented by data object dsd. This is a symmetric matrix where the (i, j)-th entry is the intersection number Di.Dj, where the Di are the irreducible divisors. The most work involved in this is computing the self-intersections, which can be rather slow. The intersections between different divisors are generally computed very quickly. Parameter getselfints can be set to false to only compute the non-self intersections. In that case, the matrix is returned containing 0s for the diagonal entries. This should cause no confusion as all genuine self-intersections are negative numbers. The result is cached in dsd for future use. If the result has been cached without self-intersections and the intrinsic is called again with getselfints set to true, only the diagonal entries are computed.
SetVerbose("MultsAndInts", n): Maximum: 1
These intrinsics give the multiplicities of the irreducible blow-up divisors above singular points in the pullback of a divisor on S to the desingularisation. Note that the usual requirement of S to be normal here for divisor computations to be correct is not needed because we are effectively working on the desingularisation.The first two intrinsics take the singular surface S as a first argument and the divisor D as second. In the second intrinsic -- for effective divisors -- the divisor is passed in as a subscheme of S. The return value is a sequence of sequences: one sequence for the divisors above each of the singular points, which correspond 1-1 with the DesingData objects in the resolution list of S and are taken in the order of that list.
The third intrinsic returns a single sequence containing only the multiplicities above a particular singular point of S, which is represented by the DesingData object dsd passed in as second argument. The third argument is an effective divisor D given as a subscheme.
SetVerbose("MultsAndInts", n): Maximum: 1
The intrinsics are the exact analogues of the Multiplicities ones except that they also compute and return the intersection of the strict transform of divisor D with the irreducible blow-up divisors. The strict transform of D is the pullback of D to the desingularisation with all blow-up divisor components removed. The return values are in the same format as for Multiplicities except that there are now two sequences of sequences or two sequences returned. The first contains the multiplicities and the second contains the intersections with the strict transform.This can often be much slower than Intersections. Most of the work usually goes into the computation of the strict transform on the blow-up surface patches, where removing blow-up divisor components can be a heavy task.
For two effective divisors, D1 and D2, on (singular) surface S, given as subschemes of S, computes the intersection number tilde(D1).tilde(D2) of the strict transforms of the Di on the blow-up desingularisation of S. Currently, S must lie in ordinary projective space and have at worst point singularities.
For an effective divisor, D, on (singular) surface S, given as a subscheme of S, computes the self-intersection number tilde(D).tilde(D) of the strict transform of D on the blow-up desingularisation of S. Currently, S must lie in ordinary projective space and have at worst point singularities.
CheckB: BoolElt Default: true
The surface S is a surface with only point singularities and B is a sequence of polynomials on the ambient of S which are homogeneous of the same multi-degree with respect to all gradings of S and can be thought of as sections (not necessarily linearly-independent) of a linear system on S given by the vector space V of appropriately-graded polynomials generated by B.The sequence ms should be a sequence of sequences of integers giving some multiplicities for the irreducible blow-up divisors. This should be in the same format (in terms of grouping and order of divisors) as the return value of Multiplicities above.
The intrinsic computes the sublinear system of V consisting of sections whose pull-back to the blow-up desingularisation vanish on each blow-up divisor to at least the multiplicity given in ms. Note that negative and zero multiplicities have no effect. The return value is a basis of sections of this sublinear system, which will be linearly-independent modulo the defining ideal of S.
The implementation initially checks B for linear-independence modulo the ideal of S and, if necessary, replaces it with a linearly-independent subsequence that generates the same V. To save a little time, the user can set parameter CheckB to false when it is known that B gives linearly-independent sections and then this initial check is skipped.
One intended use of this intrinsic is to compute Riemann-Roch spaces of strict transforms of divisors on S. The second and third long examples at the end of this section give illustrations of this. We briefly indicate how it works generally, with D a divisor on S and tilde(D), its strict transform.
Assume that the Riemann-Roch space of D has been computed, using the sheaf or divisor functionality for example, on S. Let it be given by a basis of polynomials B1 and denominator d (all polynomials having the same grading). Then the divisor of vanishing of d on S will be D + R for some effective divisor R and B1 is the basis of polynomials (of the same grading) whose divisor of zeros contains R. When we pull-back to the desingularisation of S, the divisor of d pulls back to tilde(D) + tilde(R) + ∑i ai Di where Di runs over the irreducible blow-up divisors and ai is the multiplicity of Di in the pull back of the divisor of d. It is easy to see that the Riemann-Roch space of tilde(D) is the space of rational functions on S with basis f/d where f runs through a basis of the sublinear system of the one generated by B1 that consists of sections that vanish on each Di to multiplicity at least ai, i.e. the multiplicity of d.
Warning: It is important to remember that if D1 lies in the rational equivalence class of D on S, its strict transform will not generally be rationally equivalent to tilde(D) on the desingularisation because the two will differ by a non-principal divisor supported on blow-up divisors. So, it is important to make sure that the call to get the Riemann-Roch space on S is applied to the correct divisor. This will be seen in the third example below, blowing down lines on a degenerate Del Pezzo surface. There, a hyperplane divisor has to be added to the sum of the divisors of two lines. The hyperplane divisor has to avoid the singular points for the strict transform of the resulting divisor to be in the class that we want on the desingularised Del Pezzo.
This subsection deals with blow-up resolution intrinsics that involves canonical divisors on the desingularisation. They are available for surfaces in ordinary projective space Pn of any dimension (as of Magma V2.25), with a few specific to P3. Along with the intrinsics listed here, there are also important intrinsics for birational invariants (GeometricGenusOfDesingularization, ArithmeticGenusOfDesingularization and PlurigenusOfDesingularization) and for adjoint maps (HomAdjoints), which include pluri-canonical and pluri-anticanonical maps, that are available for resolutions of singularities of blow-up or formal type and are described in subsection Adjoint Systems and Birational Invariants of the next section on hypersurfaces.
This is the fundamental intrinsic to get information about divisor multiplicities of irreducible blow-up divisors of a blow-up desingularisation in the divisors of canonical differential forms (sections of the canonical sheaf: differentials of type (2,0)) when argument dsd is a desingularisation data object for a hypersurface S. Let F be the homogeneous polynomial of degree d in 4 variables defining the singular surface S in P3.Let ω be the `natural' pseudo-differential form which is given by dx^ dy/(∂f/ ∂z) = dy^ dz/(∂f/∂x) = dz^ dx/(∂f/∂y) on a standard affine patch of S with affine coordinates x, y, z where f is the defining polynomial of S on the patch: the dehomogenisation of F at the variable which is set to 1. With an appropriate choice of signs between the standard patches, ∓ this form multiplied by a homogeneous polynomial G of degree d - 4 gives a genuine global differential form on S, ωG. That is, the ∓ω form on the standard patches multiplied by the dehomogenisation of G on those patches gives a set of patch forms that agree on the patch intersections. On each patch, ω gives a form that has no zeros or poles outside of the singular points.
The intrinsic returns the sequence of multiplicities of the irreducible blow-up divisors which lie above the singular point p in the divisor of the pull-back of ω. Here p is the singular point represented by desingularisation object dsd.
The pull-back of genuine differential form ωG has divisor equal to the (pull-back of the) effective divisor of G added to the divisor supported on all blow-up components (over all singular points) with multiplicities given by the return values of this intrinsic for the various dsds.
Note that all of these multiplicities are ≤0 and that they all equal 0 over a singular point p if and only if p is a simple singularity. This is why simple singularities can be ignored for most canonical divisor questions.
Examples of the use of DifferentialMultiplicities are given in the first two long examples at the end of the section. It is also used by many of the other package intrinsics relating to the canonical divisor.
UseProj: BoolElt Default: true
Computes a basis for everywhere defined differential 2-forms [elements of ^2Ω1(S)] on the desingularisation of surface S, that lies in ordinary projective space.The third return value is a sequence [f1, f2, f3] representing a base meromorphic 2-form w. Here the fi are rational functions on S given as quotients of homogeneous polynomials on the ambient of S of the same degree and f3 is usually 1. w = f3 * (d(f1) ^ d(f2)).
The first 2 return values are a sequence of homogeneous polynomials [b1, ..., bn] and a homogeneous polynomial F (all in the coordinate ring of the ambient of S), all of the same degree d, such that a basis of holomorphic two-forms is given by the 2-forms (bi/F) * w. If the space of holomorphic two forms only contains 0, B will be the empty sequence.
The fourth return value is a data list giving the divisor of w on the desingularisation of S. This can be used in more general computations for S ⊂Pn, n ≥4 in the same way that DifferentialMultiplicities is used for hypersurfaces in P3.
If S lies in Pn, n ≥6, then the intrinsic tries to simplify the differential computations by first projecting the surface isomorphically down to one in P5, performing the computation of the divisor of a meromorphic 2-form there, and pulling back the result. This may actually be slower, though, as the equations for the surface may become messier. To avoid this projection method, the UseProj parameter can be set to false. Its default is true.
Once computed, the data for the base 2-form is cached for use in other intrinsics.
UseProj: BoolElt Default: true
For ordinary projective surface S, with only point singularities if S is not a hypersurface, returns a (possibly empty) sequence of homogeneous polynomials on the ambient of S that give the mth pluricanonical map on the blow-up desingularisation X of S for m ∈Z [The length of the sequence is the dimension of H0(X, (^2Ω1(X)) tensor m)]. If the ambient of S is Pn, n ≥4, this uses the base meromorphic 2-form and its divisor as computed by the previous intrinsic. If that hasn't already been computed and cached then it is computed first and the UseProj parameter has the same meaning as before. This intrinsic is essentially equivalent to HomAdjoints(S,m,0).
> P4<x,y,z,t,u> := ProjectiveSpace(Rationals(),4); > S := Surface(P4,[z^2+x*t-y*t-z*u, x*z-y*z+x*u]); > dsds := ResolveSingularSurface(S); > #dsds; 2 > B,F,w,div_dat := BasisOfHolomorphicTwoForms(S); > B; F; [] u^2Here there are no everywhere-defined 2-forms (as expected on a rational surface) as follows from the empty B, but we can also examine the base meromorphic 2-form and its divisor.
> w;
[
$.1/$.2,
$.5/$.2,
1
]
The 2-form w is dx1^ dx4 on the second affine patch.
> div_dat;
[*
[
<Scheme over Rational Field defined by
x^2*t - 2*x*y*t + y^2*t - y*z*u + 2*x*u^2,
x*z - y*z + x*u,
z^2 + x*t - y*t - z*u,
x - y,
z,
u, 4>,
<Scheme over Rational Field defined by
x*t + 2*u^2,
y,
z + u, -3>,
<Scheme over Rational Field defined by
x,
y,
z - u, -1>,
<Scheme over Rational Field defined by
x,
y,
z, -1>
],
<[*
Object containing data about the exceptional divisors over singularity
Scheme over Rational Field defined by
$.1 - 1,
$.2,
$.3,
$.4
of algebraic surface
Surface over Rational Field defined by
$.1*$.3 + $.2^2 - $.2*$.4 - $.3,
$.1*$.2 + $.1*$.4 - $.2,
Object containing data about the exceptional divisors over singularity
Scheme over Rational Field defined by
$.1,
$.2,
$.3,
$.4
of algebraic surface
Surface over Rational Field defined by
$.1 - $.2 + $.3^2 - $.3*$.4,
$.1*$.3 + $.1*$.4 - $.2*$.3
*], [
[ 2 ],
[ -2, 1 ]
]>
*]
w vanishes to order 4 on (the strict transform of) a line [(-1)-curve] on S
and has poles at a conic (to order 3) and two other line components (to order 1)
of the y=0 hyperplane divisor of S.
It vanishes to order 2 on the single irreducible blow-up divisor over the singular
point represented by dsds[1] and has a pole of order 2 and zero of order 1 respectively
at the two irreducible blow-up divisors over the other singular point, represented by
dsds[2].
We check that the (-1)-pluricanonical map is just the inclusion map of S into P4 up to a linear change of coordinates. This is because S is the anticanonical embedding of the desingularised weak Del Pezzo.
> B := PluriCanonicalBasis(S,-1);
> B;
[
x*u - 2/3*y*u,
y*z - 2*y*u,
z^2 - 4*u^2,
z*t - 2*t*u,
z*u - 2*u^2
]
> // All elements of B are divisible by z-2u except the first.
> // Change B[1] by an element of Ideal(S) to also give the divisibility.
> B[1] := B[1]-(1/3)*(x*z - y*z + x*u);
> B;
[
-1/3*x*z + 1/3*y*z + 2/3*x*u - 2/3*y*u,
y*z - 2*y*u,
z^2 - 4*u^2,
z*t - 2*t*u,
z*u - 2*u^2
]
> [b div (z-2*u) : b in B];
[
-1/3*x + 1/3*y,
y,
z + 2*u,
t,
u
]
Returns the self-intersection of the canonical divisor, K.K or K2, on the blow-up desingularisation of surface S. Again, S must be an ordinary projective surface with only point singularities. If the desingularisation (by blow-up) of S hasn't already been cached, it is computed and cached at the start.Note that this is not a birational invariant of desingularisations. It depends on the desingularisation that magma computes. It gives a measure of how non-minimal the desingularisation is.
SetVerbose("MultsAndInts", n): Maximum: 1
The first two intrinsics return the intersection number tilde(D).K of the strict transform tilde(D) of divisor D with the canonical divisor on the blow-up desingularisation of surface ordinary projective surface S. Recall that the strict transform of a divisor is the pull-back of the divisor to the blow-up with all blow-up divisor components removed (equivalently, the Zariski closure in the desingularisation of the divisor with the singular points of S removed). The second intrinsic is for an effective divisor which is given as a subscheme of S.The third intrinsic return the intersection number with the canonical divisor of the i-th irreducible blow-up divisor over the surface singular point p that is represented by desingularisation data object dsd.
SetVerbose("UseSheaf", n): Maximum: true
SetVerbose("Kodim", n): Maximum: 10
For an ordinary projective surface, S, with only point singularities, and of Kodaira dimension ≥0 (equivalently, S is not rational or birationally ruled), returns two lists, the first containing all of the (-1)-curves on the blow-up desingularisation X of S (including irreducible divisors that split into a union of (-1)-curves over the algebraic closure of the base field), and the second containing all other irreducible curves on X that are contracted to points in the map from X to its minimal model.Each of the two lists returned is itself a list containing two sequences. The first sequence consists of curves that are strict transforms of irreducible curves D on S. These are just returned as the subschemes of S equal to D. The second sequence consists of irreducible blow-up divisors. These are given as sequence pairs [i, j], representing the jth irreducible divisor of the ith desingularisation data structure dsd = dsds[i] of S, where dsds is the desingularisation list returned by ResolveSingularSurface. (See example below and also the first example of the extended examples subsection).
The method is based upon finding all irreducible divisors common to a basis of global sections of a sufficiently large multiple of the canonical divisor. When S is singular, this is based upon computations using the base meromorphic divisor returned by BasisOfHolomorphicTwoForms. When S is already non-singular it can be quicker to use the canonical sheaf. To use this sheaf in the non-singular case, the parameter UseSheaf should be set to true (the default).
The smallest multiple of the canonical divisor required and some other issues require that the exact Kodaira dimension of S is known. If this is known to the user, and not necessarily already cached in the surface data, they can enter it by setting the Kodim parameter to a value between 0 and 2 inclusive. Any other value is ignored and the Kodaira dimension is then computed at the start.
> P<a,b,c,d> := ProjectiveSpace(Rationals(),3);
> S := Surface(P, a^6+2*a^5*b-2*a^4*b^2+2*a^2*b^4+b^6+a^4*d^2+2*a^3*b*d^2+
> a^2*b^2*d^2-a*b^3*d^2-2*b^4*d^2-a^2*b*d^3-2*a*b^2*d^3+2*a^2*d^4+b^2*d^4-
> c^2*d^4-a*d^5);
> dsds := ResolveSingularSurface(S);
> #dsds; // only one singular point at [0,0,1,0]
1
> l1,l2 := ExceptionalDivisors(S);
> l1;
[*
[
Scheme over Rational Field defined by
a^5 + a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4 + b^5,
d,
Scheme over Rational Field defined by
a + b,
d
],
[]
*]
> l2;
[*
[],
[
[ 1, 1 ],
[ 1, 2 ]
]
*]
So there are 6 (-1)-curves on the desingularisation X of S : 5 Galois-conjugate ones
given by the first entry in l1[1], which is easily seen to split into 5 distinct lines
over an extension of Q, and one defined over Q, given by the line l1[1][2].
There are no blow-up (-1)-curves. On the other hand l2 shows that, in minimising X, there are
irreducible blow-up divisors that are also contracted. These consist of the first and
second of the three irreducible blow-up divisors over the singularity represented by the
first (and only!) desingularisation data object dsds[1]. Further analysis (see extended
example 2 below) shows that the first irreducible blow-up divisor splits into 5
conjugate (-2)-curves and the second is a (-2)-curve defined over Q.
For a blow-up desingularisation of a surface, functionality is now provided (as of magma V2.25) to produce quite complete information on the configuration of exceptional divisors over a singular cluster-point, along with additional data about the divisors, in the form of a dual resolution graph. Using this data, we have also added intrinsics to compute the total number of points on the "blow-up fibre" over a singular point or the zeta-function of this fibre, for surfaces over finite fields. This is useful for point-counting on the desingularised surface. There are also versions of the intrinsics to compute the graphs and perform the point-counting for the minimal blow-up resolution, which is the initial blow-up resolution with any (-1)-curves amongst the exceptional divisors iteratively blown back down until no more remain.
The dual resolution graph of a blow-up desingularisation over a singular point p of a surface, represented by the dsd desingularisation data object argument, is a multi-edge graph in which the nodes represent the irreducible divisors lying over p and edges represent intersections between these divisors. In addition, singular points of a divisor are represented by edge-loops from the corresponding node to itself. The graph, along with associated data about the divisors in node and edge labels (to be described below), is computed and returned by this intrinsic. It is of magma's MultiGraph type. Once computed, it is stored for immediate return by later calls to the intrinsic.There are two further return values that completely specify the intersection configuration. The first is a sequence of pairs of edges giving singular intersections: intersections between divisors where the intersection point is a singular point on (at least) one of the divisors. The corresponding entry for this would be a sequence pair [e, f] where e is the graph loop-edge representing the singular point and f is the non-loop edge between the divisor nodes representing the intersection. If there was an intersection between divisors where the intersection point was a singular point on both divisors, there would be two entries in the sequence: [e1, f] and [e2, f]. Similarly, multiple intersections involving one or more singular points would give several entries in the obvious way.
The other sequence contains multiple-intersections, i.e. intersection points that lie on 3 or more blow-up divisors. The entry for a multiple-intersection is a sequence containing each edge that represents the intersection between 2 divisors at that point. Thus, if n ≥3 divisors meet at the point, the entry will be a sequence containing n(n - 1)/2 graph edges.
Note that both of these sequences will be empty in the vast majority of cases.
Labels: The labels of the edges and nodes of the graph give information about the isomorphism types of the divisors and their canonical intersections and also relates the intersection points to the local blow-up affine patches accessible through dsd. This information is largely available through the accessor functions below, but it may be useful to the user to obtain it directly through the graph labels.
Node labels: the label for a node representing irreducible blow-up divisor D, is a list [* s1,s2,s3,s4 *] containing three elements.
Edge labels 1: the label for a non-loop edge, representing an intersection p between blow-up divisors D1 and D2 is a list [* s1,s2,s3,s4 *] containing 4 elements.
- s1 is the self-intersection number of D, D.D
- s2 is the function field (type FldFunFracSch) of D [computed from a blow-up affine patch of D], which determines the curve normalisation of D
- s3 is the canonical intersection number of D, D.K
- s4 is a list of three elements [ * pi, ps, pls * ]. pi is the blow-up patch index of dsd which gives the representative affine patch of D from which s2 is computed. The other two elements give the finite set of scheme-theoretic points of D off this patch. ps contains a list of representatives for these points on other blow-up patches in the form <p, j> where p is the cluster giving the point on blow-up patch index j of dsd. pls is a sequence of places of function field s2 corresponding to these points. [see examples below]
Edge labels 2: the label for a loop edge, representing a singular point p on blow-up divisor D is a list [* s1,s2,s3 *] containing 3 elements.
- s1 is the local intersection number (D1.D2)p
- s2 is a sequence of 3 indices [i, j1, j2]. The meaning is that p is a (non-singular) point on the blow-up surface patch index i of dsd and that j1, j2 are the indices in the sequence of blow-up divisors for patch i of D1, D2 [see examples below]
- s3 is p as a cluster on the blow-up surface patch index i.
- s4 is a 2-element list [ * l1, l2 * ]. li is a sequence containing all of the places of the function field of Di over the point p. If p is non-singular on Di, li will only contain one place. This determines p "globally" on Di.
- s1 is a cluster p on one of the blow-up patches of dsd giving the singular point
- s2 is an integer i, giving the index of the blow-up patch that p is on.
- s3 is a sequence of all of the places of the function field of D over p.
This is analogous to DualResolutionGraph, but gives the graph for the minimal resolution over singular point p of a surface represented by the dsd desingularisation data object argument. The minimal resolution is the initial (blow-up) resolution followed by the iterative contraction (blowing-down) of any (-1)-curves amongst the irreducible blow-up exceptional curves, until no more exist. The minimal resolution is "canonically" associated to the singularity and does not depend on the initial method of resolution (which is by iterated blow-up here).Note that blowing down a (-1)-curve increases the self-intersection number of any curve intersecting it, so the minimisation process doesn't just involve (Galois-conjugate sets of) curves with self-intersection -1 in the initial resolution. For example, if a (-1)-curve C1 intersects a (-2)-curve C2 transversely in a single point, then, after blowing down C1, C2 becomes a new (-1)-curve.
The return values are the same as for DualResolutionGraph with an additional fourth return value, which is the intersection matrix of the irreducible exceptional divisors in the minimal resolution, taken in the order in which they appear as nodes in the minimal graph.
The node and edge labels in the minimal graph are the same as for the full dual resolution graph except with small changes for edges representing divisor singularities and intersections that are contractions of divisors in the minimisation process. For these, the cluster on a blow-up patch representing the singularity/intersection point is replaced by a string contn where n=1, 2, ... is the stage of contractions at which the point is produced (in the minimisation process, a maximal disjoint set of (-1)-curves is considered to be contracted at each stage).
In practice, the original blow-up resolution in already minimal in the vast majority of cases.
> P<x,y,z,t> := ProjectiveSpace(Rationals(),3);
> S := Surface(P,
> z^2*t^7-x^8*t-8*x^7*t*y-4*x^6*t^2*y-24*x^6*t*y^2-24*x^5*t^2*y^2-
> 4*x^6*y^3-32*x^5*t*y^3-52*x^4*t^2*y^3-24*x^5*y^4-32*x^4*t*y^4-
> 48*x^3*t^2*y^4-52*x^4*y^5-64*x^3*t*y^5-16*x^2*t^2*y^5-48*x^3*y^6-
> 96*x^2*t*y^6-16*x^2*y^7-64*x*t*y^7-16*t*y^8);
> dsds,typ := ResolveSingularSurface(S);
> typ; // resolution of blow-up type
2
> #dsds; // 5 Galois-conjugate sets of singular points
5
> dsd := dsds[4];
> IntersectionMatrix(dsd);
[-2 0 0 0 0 0 0 1 0 0 0]
[ 0 -2 0 0 0 0 0 1 0 0 0]
[ 0 0 -2 0 0 0 0 0 1 0 0]
[ 0 0 0 -2 0 0 0 0 1 0 0]
[ 0 0 0 0 -2 0 0 0 0 0 1]
[ 0 0 0 0 0 -2 0 0 0 1 0]
[ 0 0 0 0 0 0 -2 0 0 1 0]
[ 1 1 0 0 0 0 0 -2 0 0 1]
[ 0 0 1 1 0 0 0 0 -2 0 1]
[ 0 0 0 0 0 1 1 0 0 -2 1]
[ 0 0 0 0 1 0 0 1 1 1 -4]
> G,sngs,multi_ints := DualResolutionGraph(dsd);
> G;
Multigraph
Vertex Neighbours
1 8 ;
2 8 ;
3 9 ;
4 9 ;
5 11 ;
6 10 ;
7 10 ;
8 11 2 1 ;
9 11 4 3 ;
10 11 7 6 ;
11 10 9 8 5 ;
> sngs; multi_ints;
[]
[]
> vG := VertexSet(G);
> Label(vG.11);
[* -4,
Function Field of Curve over Rational Field defined by
$.2,
$.3,
2, [* 6, [*
<Scheme over Rational Field defined by
$.1 + 1/2,
$.2,
$.3, 10>,
<Scheme over Rational Field defined by
$.1 + 1,
$.2,
$.3, 13>,
<Scheme over Rational Field defined by
$.1,
$.2,
$.3, 16>
*], [*
[
Place at (-1/2 : 0 : 0 : 1)
],
[
Place at (-1 : 0 : 0 : 1)
],
[
Place at (1 : 0 : 0 : 0)
]
*] *]
*]
> Edges(vG.11,vG.11);
[]
The above shows that the 11-th blow-up divisor D is a non-singular (no edges to itself) rational
curve (from the function field) with self-intersection -4 and canonical intersection 2.
It's representative affine patch is the 6-th one of the blow-up patches of the dsd
resolution and there are three (rational) points not on that patch, with chosen representatives
on the 10-th, 13-th and 16-th blow-up patches of dsd and corresponding places of its
function field as listed. We can get some of this data using intrinsics described below
> SelfIntersection(G,11); -4 > CanonicalIntersection(G,11); 2 > ArithmeticGenus(G,11); 0We can use the dsd accessor intrinsics to look at the actual representative affine patch, Da, of D
> dat11 := [e : e in BlowUpDivisorAllPatches(dsd,11) | e[4] eq 6][1]; > Da := dat11[1]; Da; Curve over Rational Field defined by 1/4*$.2^2 - 1/4*$.3^2 + $.2, $.2, $.3 > Ambient(Da); Affine Space of dimension 3 over Rational Field Variables: $.1, $.2, $.3Da is a line in affine 3-space and the function field of D has the usual projective closure of Da as 'its curve'. The first 2 of the list of 3 places for points of D off the representative patch appear to actually lie on Da. These can only be illegitimate representations of points on the blow-up divisor D because they are singular points on the 6th surface blow-up patch. This is easily checked:
> Xa6 := dat11[2]; // the 6th surface blow-up patch > Xa6; Scheme over Rational Field defined by $.1^8*$.2^3 + $.1^8*$.2^2 + 3*$.1^7*$.2^3 + 4*$.1^7*$.2^2 + 13/4*$.1^6*$.2^3 + 6*$.1^6*$.2^2 + 3/2*$.1^5*$.2^3 + 4*$.1^5*$.2^2 + 1/4*$.1^4*$.2^3 + 2*$.1^4*$.2^2 + $.1^4*$.2 + 2*$.1^3*$.2^2 + 3*$.1^3*$.2 + 3/2*$.1^2*$.2^2 + 13/4*$.1^2*$.2 + 1/2*$.1*$.2^2 + 3/2*$.1*$.2 + 1/16*$.2^2 - 1/16*$.3^2 + 1/4*$.2 > IsSingular(Xa6![-1/2,0,0]); IsSingular(Xa6![-1,0,0]); true (-1/2, 0, 0) true (-1, 0, 0)From the graph, D intersects the 5th, 8th, 9th and 10th blow-up divisors. We look at its intersection(s) with the 10th.
> es := Edges(vG.10,vG.11); es; // single intersection point
[ < {10, 11}, 19 > ]
> e := es[1];
> Label(e);
[* 1,
[ 16, 2, 1 ],
Scheme over Rational Field defined by
$.1,
$.2,
$.3,
[*
[
Place at (0 : 1 : 0 : 0)
],
[
Place at (1 : 0 : 0 : 0)
]
*]
*]
> // Local intersection multiplicity is 1 (its a transverse intersection).
> // Can also get this by
> LocalIntersectionNumber(e);
1
> A<a,b,c> := AffineSpace(Rationals(),3);
> S := Surface(A,-a^6-6*a^5*b-12*a^4*b^2-2*a^4*c-8*a^3*b^3-
> 8*a^3*b*c-8*a^2*b^2*c+a^2*b^2-a^2*c^2+2*a*b^3-2*a*b*c^2+
> 2*a*b*c+b^2*c+c^2);
> dsds := ResolveSingularSurface(S);
> #dsds;
1
> dsd := dsds[1];
> IntersectionMatrix(dsd);
[-2 1 1]
[ 1 -2 1]
[ 1 1 -4]
> G,sngs,multi_ints := DualResolutionGraph(dsd);
> G;
Multigraph
Vertex Neighbours
1 3 2 ;
2 3 1 ;
3 2 1 ;
> sngs;
[]
> multi_ints;
[
[ < {1, 2}, 1 >, < {1, 3}, 3 >, < {2, 3}, 5 > ]
]
The graph (or intersection matrix) shows that each blow-up divisor
meets each of the other 2, but the only way (beyond detailed study of the
graph labels) to distinguish between
a triangular configuration, where the three divisors don't have a
common point, and the actual one, with a single intersection point
between the 3, is via the multi_ints return value.
The self-intersection number D.D of the ith irreducible blow-up divisor over the surface singularity whose blow-up resolution is represented by desingularisation data object dsd.In the second version, the dual resolution graph G of the desingularisation is given as argument rather than dsd. In this version, G may also be the minimal dual resolution graph, in which case i refers to the ith divisor (node) of that graph.
The canonical intersection number, D.K, of the ith irreducible blow-up divisor over the surface singularity whose blow-up resolution is represented by desingularisation data object dsd with the canonical divisor K of the desingularised surface.In the second version, the dual resolution graph G of the desingularisation is given as argument rather than dsd. In this version, G may also be the minimal dual resolution graph, in which case i refers to the ith divisor (node) of that graph.
The arithmetic genus of the ith irreducible blow-up divisor over the surface singularity whose blow-up resolution is represented by desingularisation data object dsd.In the second version, the dual resolution graph G of the desingularisation is given as argument rather than dsd. In this version, G may also be the minimal dual resolution graph, in which case i refers to the ith divisor (node) of that graph.
The genus of the normalisation of the ith irreducible blow-up divisor over the surface singularity whose blow-up resolution is represented by desingularisation data object dsd.In the second version, the dual resolution graph G of the desingularisation is given as argument rather than dsd. In this version, G may also be the minimal dual resolution graph, in which case i refers to the ith divisor (node) of that graph.
The argument e should be the edge in a (minimal) dual resolution graph joining two different nodes, representing an intersection point p between irreducible exceptional divisors D and E. Return the local intersection number at p, (D.E)p. (For two effective surface divisors D, E with no common component, D.E is the sum over all intersection points of the local intersection numbers).
For the blow-up resolution of a singular (scheme-theoretic) point p corresponding to desingularisation data object dsd on a surface over a finite field F, returns the number of F-rational points on the (reduced) fibre over p, which is the union of reduced exceptional divisors over p. Note that if p is not F-rational, this is 0. Uses the dual resolution graph of dsd.
This is similar to the intrinsic NumberOfPointsOnResolutionFibre above, except it computes the number of points on the fibre of the minimal resolution over singular point p to which dsd corresponds. The minimal dual resolution graph is used.
With the same assumptions as for NumberOfPointsOnResolutionFibre, this intrinsic returns the zeta-function of the reduced resolution fibre over the singular point to which dsd corresponds. Again, the dual resolution graph of dsd is used.
With the same assumptions as for NumberOfPointsOnResolutionFibre, returns the zeta-function of the reduced resolution fibre over the singular point to which dsd corresponds. Again, uses the dual resolution graph.
As above, except computes the zeta-function of the reduced fibre of the minimal resolution over singular point p to which dsd corresponds. Uses the minimal dual resolution graph.
> P<x,y,z,t> := ProjectiveSpace(GF(37),3);
> S := Surface(P,
> z^2*t^7-x^8*t-8*x^7*t*y-4*x^6*t^2*y-24*x^6*t*y^2-24*x^5*t^2*y^2-
> 4*x^6*y^3-32*x^5*t*y^3-52*x^4*t^2*y^3-24*x^5*y^4-32*x^4*t*y^4-
> 48*x^3*t^2*y^4-52*x^4*y^5-64*x^3*t*y^5-16*x^2*t^2*y^5-48*x^3*y^6-
> 96*x^2*t*y^6-16*x^2*y^7-64*x*t*y^7-16*t*y^8);
> dsds := ResolveSingularSurface(S);
> // one of the pairs of conjugate simple singularities has split over GF(37)
> #dsds;
6
> [NumberOfBlowUpDivisors(dsd) : dsd in dsds];
[ 1, 1, 3, 3, 11, 38 ]
> dsd := dsds[5];
> ZetaFunctionOfResolutionFibre(dsd);
1/(177917621779460413*t^12 - 230812049876056752*t^11 + 60042323785325574*t^10 -
7727454798626200*t^9 + 610886629350855*t^8 - 32512885054848*t^7 +
1217402509092*t^6 - 32655381264*t^5 + 626830875*t^4 - 8433040*t^3 +
75702*t^2 - 408*t + 1)
> Factorisation(Denominator($1));
[
<$.1 - 1, 1>,
<37*$.1 - 1, 11>
]
This is easily seen to be the zeta-function of 11 projective lines over GF(37)
with 10 intersection points.
> NumberOfPointsOnResolutionFibre(dsd);
408
> A<a,b,c> := AffineSpace(GF(37),3);
> S := Surface(A,-a^6-6*a^5*b-12*a^4*b^2-2*a^4*c-8*a^3*b^3-
> 8*a^3*b*c-8*a^2*b^2*c+a^2*b^2-a^2*c^2+2*a*b^3-2*a*b*c^2+
> 2*a*b*c+b^2*c+c^2);
> dsds := ResolveSingularSurface(S);
> #dsds;
> dsd := dsds[1];
> ZetaFunctionOfResolutionFibre(dsd);
1/(50653*t^4 - 54760*t^3 + 4218*t^2 - 112*t + 1)
> Factorisation(Denominator($1));
[
<$.1 - 1, 1>,
<37*$.1 - 1, 3>
]
> NumberOfPointsOnResolutionFibre(dsd);
112
We show how we can quickly compute all this with the surface intrinsics, along with the intersection matrices of the curves (which give the intersection graphs) over each conjugate set of singular points.
> P<x,y,z,t> := ProjectiveSpace(Rationals(),3); > S := Surface(P, > z^2*t^7-x^8*t-8*x^7*t*y-4*x^6*t^2*y-24*x^6*t*y^2-24*x^5*t^2*y^2- > 4*x^6*y^3-32*x^5*t*y^3-52*x^4*t^2*y^3-24*x^5*y^4-32*x^4*t*y^4- > 48*x^3*t^2*y^4-52*x^4*y^5-64*x^3*t*y^5-16*x^2*t^2*y^5-48*x^3*y^6- > 96*x^2*t*y^6-16*x^2*y^7-64*x*t*y^7-16*t*y^8); > dsds,typ := ResolveSingularSurface(S); > typ; // resolution of blow-up type 2 > #dsds; // 5 conjugacy classes of singular points 5 > IntersectionMatrix(dsds[1]); // 2 conjugate A3 singularities [-4 0 2] [ 0 -4 2] [ 2 2 -4] > IntersectionMatrix(dsds[2]); // 2 conjugate A1 singularities [-4] > IntersectionMatrix(dsds[3]); // 2 more conjugate A1 singularities [-4] > IntersectionMatrix(dsds[4]); // 11-component singular point [-2 0 0 0 0 0 0 1 0 0 0] [ 0 -2 0 0 0 0 0 1 0 0 0] [ 0 0 -2 0 0 0 0 0 1 0 0] [ 0 0 0 -2 0 0 0 0 1 0 0] [ 0 0 0 0 -2 0 0 0 0 0 1] [ 0 0 0 0 0 -2 0 0 0 1 0] [ 0 0 0 0 0 0 -2 0 0 1 0] [ 1 1 0 0 0 0 0 -2 0 0 1] [ 0 0 1 1 0 0 0 0 -2 0 1] [ 0 0 0 0 0 1 1 0 0 -2 1] [ 0 0 0 0 1 0 0 1 1 1 -4]Computing the self-intersections for the 38 components over the final singularity takes about five minutes, so we don't include it here. The computation of the intersections between different components is still very fast.
> // 38-component singular point > IntersectionMatrix(dsds[5] : get_self_ints := false); [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1] [1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0] > GeometricGenusOfDesingularization(S); 3A sequence of degree 5 polynomials defining (rationally) the canonical map is returned by HomAdjoints(1,0,S). For simplicity, we will then just write down an equivalent map to P1, which is obvious from inspection, factoring out the common t3 and combining with the standard birational projection of the image (ac=b2 in P2(a, b, c)) to P1. We then check that this is an elliptic fibration map by looking at a particular fibre, which is an irreducible genus one curve.
> HomAdjoints(1,0,S);
[
x^2*t^3,
x*y*t^3,
y^2*t^3
]
> P1<u,v> := Curve(ProjectiveSpace(Rationals(),1));
> fib_map := map<S->P1|[x,y]>;
> bs := ReducedSubscheme(BaseScheme(fib_map));
> bs;
Scheme over Rational Field defined by
z*t,
x,
y
> // bs is just the two rational singular points
> f1 := Complement((P1![1,1])@@fib_map,bs);
> GroebnerBasis(f1);
[
y^9 + 337/144*y^8*t + y^7*t^2 - 1/144*z^2*t^7,
x - y
]
> IsIrreducible(f1); IsReduced(f1);
true
true
> Genus(Curve(f1));
1
We can go further here and find out exactly the structure of bad fibres
of the minimal model of X and how they relate to the graphs of exceptional
divisors over the singular points. Firstly, we check that our desingularisation
is not minimal by computing its first Chern class. This is the self-intersection
number K2 where K is a canonical divisor on the desingularisation. It turns
out to be -27. As K2 for the minimal model is 0, the desingularisation is
the minimal model blown up 27 times. Normally, the intrinsic
FirstChernClassOfDesingularization would be called here. As that uses
full intersection matrices for the exceptional blow-up divisors and we have
avoided computing the fifth one fully to save time, we cheat by manually
inserting the diagonal entries of the fifth matrix and then perform the
computation undertaken by the intrinsic directly.
> imats := [*IntersectionMatrix(dsds[i]) : i in [1..4]*]; > imat5 := IntersectionMatrix(dsds[5] : get_self_ints := false); > imat5 := imat5 + DiagonalMatrix([-2 : i in [1..33]] cat [-3,-3,-3,-4,-3]); > Append(~imats,imat5); > dms := [DifferentialMultiplicities(dsd) : dsd in dsds]; > i2 := &+[(m*imats[i]*Transpose(m))[1,1] where m is Matrix(Vector(dms[i])) : > i in [1..5]]; > d := Degree(S); // d=9 > K2 := d*(d-4)^2+i2; > K2; -27We can find the irreducible divisors that are contracted to give the minimal model by calling the ExceptionalDivisors intrinsic. This is a little slow in this complex example but still only takes about a minute and a half on our machine.
> time m1s,others := ExceptionalDivisors(S);
Time: 92.490
> m1s; // The (-1)-curves
[*
[
Scheme over Rational Field defined by
x + 2*y,
t,
Scheme over Rational Field defined by
x + y,
t,
Scheme over Rational Field defined by
x,
t,
Scheme over Rational Field defined by
y,
t
],
[]
*]
> others; // the other curves to be contracted
[*
[],
[
[ 5, 1 ],
[ 5, 2 ],
[ 5, 3 ],
[ 5, 4 ],
[ 5, 5 ],
[ 5, 6 ],
[ 5, 7 ],
[ 5, 8 ],
[ 5, 9 ],
[ 5, 10 ],
[ 5, 19 ],
[ 5, 20 ],
[ 5, 21 ],
[ 5, 22 ],
[ 5, 25 ],
[ 5, 26 ],
[ 5, 27 ],
[ 5, 28 ],
[ 5, 29 ],
[ 5, 30 ],
[ 5, 31 ],
[ 5, 32 ],
[ 5, 33 ]
]
*]
Thus, the (strict transforms of the) four lines in m1s[1] are the (-1)-curves
and those along with the 23 blow-up divisors over the fifth
singular point whose indices are given byothers[2] are the components
that need to be contracted to give the minimal model for X. This tallies with
the earlier K2 computation (and renders it redundant!).
Further analysis shows the following.
The irreducible divisors above the 2 Q-rational singular points map to one of the four points 0, - 1, - 2, ∞ on P1 under fibmap except for the curve of self-intersection -3 over singularity 4 and the curve of self-intersection -4 over singularity 5, both of which map onto P1. This is computed using BlowUpDivisors to get affine patches of the divisors along with the maps down to S.
There are four chains of curves, each starting from one of these (-1)-curves, continuing down through (-2)-curves above the fifth singularity p and stopping at the four (-3)-curves over p. The blow-down from the desingularisation to the minimal model contracts these four chains, leaving only (-2)-curves and the (-4)-curve over p. We also find another 8 irreducible curves in S over the four points of P1 mentioned above and these 8 curves are also (-2)-curves and join the remaining (non-blown-down) components over p to those over the fourth singularity. All of these (-2)-curves together - the 8 new ones and those over p and the fourth singularity, excluding the two (-4)-curves which map onto P1 - form four disjoint simple loops containing eight curves each. These are the fibres of the minimal model over the four points of P1 and are all of type I8 in the usual Neron/Kodaira notation. The curves over the six simple singular points along with an additional (-2)-curve (strict transform of a curve in S) for each singularity form the remaining bad fibres on the minimal model which give two fibres of type I4 and four of type I2.
We can deduce from all the information gained that the divisor H + K on the desingularisation (H is the hyperplane divisor and K the canonical) is base-point free and its associated map into projective space contracts precisely the curves on which all differentials vanish, so factors through the minimal model. It should give an isomorphic embedding of the minimal model into ordinary projective space. However, it maps into projective space of dimension 30, which is rather large.
We briefly illustrate a couple of these final points.
> C,_,mp := BlowUpDivisor(S,dsds[4],11); > mp := Restriction(mp,C,S : Check := false); > fmpC := Expand(mp*fib_map); > // manually remove common factors from defining equations to > // get the fibre map from C to P^1 > fmpC := map<C->P1|[e div g : e in DefiningPolynomials(fmpC)]> where > g is GCD(DefiningPolynomials(fmpC)); > C; fmpC; Curve over Rational Field defined by 1/4*$.2^2 - 1/4*$.3^2 + $.2, $.2, $.3 Mapping from: Crv: C to Crv: P1 with equations : 1 $.1 > Image(fmpC); // It's onto! Curve over Rational Field defined by 0 > C,_,mp := BlowUpDivisor(S,dsds[5],34); > mp := Restriction(mp,C,S : Check := false); > fmpC := Expand(mp*fib_map); > fmpC := map<C->P1|[e div g : e in DefiningPolynomials(fmpC)]> where > g is GCD(DefiningPolynomials(fmpC)); > C; fmpC; Curve over Rational Field defined by $.2, $.3 Mapping from: Crv: C to Crv: P1 with equations : 1 $.1*$.2 + 4*$.2 - 1/2 > Image(fmpC); // C maps to [-2:1] Scheme over Rational Field defined by 1/2*u + v > // |H+K| divisor map is the (1,1)-adjoint map : HomAdjoints gives > // defining polynomials > HKeqns := HomAdjoints(1,1,S); > #HKeqns; // It's into P^30 31
We will use the desingularisation machinery to analyse a singular model of such an S in P3, construct the divisor map on this model and check that the image has the predicted form. Note, generally that if the smooth sextic has equation f6(a, b, d)=0 in P2(a, b, d) then a birational model for the double cover in P3(a, b, c, d) is given by the degree 6 homogeneous equation F=f6(a, b, d) - c2d4=0 and this hypersurface is only singular at the point with homogeneous coordinates [0:0:1:0] so blow-up desingularisation may be applied. In the more general case of a sextic with simple curve singularities (which also leads to a K3 surface), there will be further singular points on the hypersurface model, but still only singular points (not curves) as long as coordinates are chosen so that none of the singularities of f6 are at infinity (i.e. on the line d=0).
Our example has
f6 = a6 + 2a5b - 2a4b2 + 2a2b4 + b6 + a4d2 + 2a3bd2 + a2b2d2 - ab3d2 - 2b4d2 - a2bd3 - 2ab2d3 + 2a2d4 + b2d4 - ad5
which is easily checked to be nonsingular and to have a=0 as a tritangent line. We will take the pullback of the line b=0 as an irreducible divisor of class B.
> P<a,b,c,d> := ProjectiveSpace(Rationals(),3); > S := Surface(P, a^6+2*a^5*b-2*a^4*b^2+2*a^2*b^4+b^6+a^4*d^2+2*a^3*b*d^2+ > a^2*b^2*d^2-a*b^3*d^2-2*b^4*d^2-a^2*b*d^3-2*a*b^2*d^3+2*a^2*d^4+b^2*d^4- > c^2*d^4-a*d^5); > dsds := ResolveSingularSurface(S); > #dsds; // only one singular point at [0,0,1,0] 1 > dsd := dsds[1]; > IntersectionMatrix(dsd); [-10 0 5] [ 0 -2 1] [ 5 1 -4]The first irreducible blow-up divisor D actually splits into 5 disjoint (-2)-curves over a finite extension of Q. We can see this by examining the equations for it on a chosen affine patch. To check that the components don't intersect on another patch, we could use BlowUpDivisorAllPatches, but it suffices to show that the intersection number K.D=0 (implying that K.Di=0 for each component : they are all conjugate and so have the same intersection number with the Q-rational class K), and that each component is a rational curve. Then, if any component were singular on another patch or the components intersected, D.D would be greater than -10. The other two classes are a further genus 0 nonsingular curve (also disjoint from D by the intersection matrix) and a non-singular genus 2 curve.
> C1 := BlowUpDivisor(S,dsd,1); C1; Curve over Rational Field defined by $.2^5 - $.2^4 + 3*$.2^3 - 3*$.2^2 + $.2 + 1, $.1 > CanonicalIntersection(dsd,1); 0 > C2 := BlowUpDivisor(S,dsd,2); C2; Curve over Rational Field defined by $.1, $.2 + 1 > C3 := BlowUpDivisor(S,dsd,3); C3; $.1^6*$.3^2 + 2*$.1^4*$.3^2 - 2*$.1^2*$.3^2 + 2*$.1*$.3^2 + $.3^2 - 1, $.2 > Genus(Curve(C3)); 2
As in the last example, we can get the geometric genus of the blow-up, a basis for holomorphic differentials and also the set of irreducible divisors on which all these differentials vanish. Comparing with the K2 intersection number verifies that the minimal K2 = 0 and that the desingularisation is a minimal model blown up twelve times. We also confirm that S is (birationally) a K3 surface by checking that the second plurigenus and the arithmetic genus are 1. These invariant values, along with pg=1, only occur for blow-ups of K3 surfaces.
> GeometricGenusOfDesingularization(S);
1
> PlurigenusOfDesingularization(S,2);
1
> ArithmeticGenusOfDesingularization(S);
1
> diffs := HomAdjoints(1,0,S); diffs;
[
d^2
]
> // Only one holomorphic differential up to a scalar.
> // Curves on which it vanishes are those in S on which
> // d vanishes and blowup divisors on which $d^2$ vanishes
> // to a higher order than the differential multiplicity.
> // This turns out to be the first 2 blow-up divisors.
> Dd := S meet Scheme(P,d);
> divs1 := IrreducibleComponents(ReducedSubscheme(Dd));
> divs1;
[
Scheme over Rational Field defined by
a^5 + a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4 + b^5,
d,
Scheme over Rational Field defined by
a + b,
d
]
> // The first divisor above splits into 5 disjoint lines
> // again over an extension of Q. We check that all 6
> // geometrically irreducible lines in divs1 have strict
> // transforms on the blow-up that are (-1)-curves by
> // computing that K.D=-1 for all of them
> CanonicalIntersection(S,divs1[1]);
-5
> CanonicalIntersection(S,divs1[2]);
-1
> DifferentialMultiplicities(dsd);
[ -3, -3, -6 ]
> [2*m : m in Multiplicities(S,dsd,Dd)];
[ 4, 4, 6 ]
> FirstChernClassOfDesingularization(S);
-12
The strict transform of each of the five conjugate lines in divs1 meets one of the five conjugate lines of the first irreducible blow-up divisor once transversally. The line over Q in divs1 meets blow-up divisor 2 once transversally. This is checked by computing intersection numbers. This shows that the blow-up process to go from the minimal K3 surface to our desingularisation is to blow up 6 points (one over Q and 5 conjugate ones over Q) and then to further blow up a point on each of the (-1)-curves arising.
> _,ints := MultiplicitiesAndIntersections(S,dsd,divs1[1]); > ints; [ 5, 0, 0 ] > _,ints := MultiplicitiesAndIntersections(S,dsd,divs1[2]); > ints; [ 0, 1, 0]
We now construct the 2B + Z1 divisor map. The method is to first get a basis for the Riemann-Roch space of the divisor using the normal call to the divisor/sheaf intrinsic. However, as explained in the description of the LinearSystemDivisorRestriction intrinsic, we then have to impose extra conditions at the blow-up divisors. If the denominator of the R-R space vanishes to given multiplicities at these divisors, we have to take the subspace of numerators that also vanish to at least these multiplicities. As it is slightly more convenient to get the R-R space as a separate basis of numerators and denominators using the sheaf intrinsic, we use that here.
> // First get B and Z1
> // Z1 is one of the components of the pullback of a=0 to S
> cmps := IrreducibleComponents(S meet Scheme(P,a));
> cmps;
[
Scheme over Rational Field defined by
b^3 - b*d^2 + c*d^2,
a,
Scheme over Rational Field defined by
b^3 - b*d^2 - c*d^2,
a
]
> Z1 := cmps[1];
> // B is the pullback of b=0 to S
> B := S meet Scheme(P,b);
> GroebnerBasis(B);
[
a^6 + a^4*d^2 + 2*a^2*d^4 - c^2*d^4 - a*d^5,
b
]
> rr,den := RiemannRochBasis(S,
> EquidimensionalPart((Ideal(B)^2*Ideal(Z1))+Ideal(S)));
> rr;
[
a^3,
a^2*b,
a^2*c,
a^2*d,
a*b^2,
a*b*c,
a*b*d,
a*c^2,
a*c*d,
a*d^2,
b^3 - b*d^2 - c*d^2
]
> den;
a*b^2
> // Do the linear system restriction at blow-up divisors
> mults := Multiplicities(S,dsd,S meet Scheme(P,den)); mults;
[3,3,6]
> div_map_eqns := LinearSystemDivisorRestriction(S,rr,[mults]:
> CheckB := false);
> div_map_eqns;
[
a^3,
a^2*b,
a^2*d,
a*b^2,
a*b*d,
a*d^2,
b^3 - b*d^2 - c*d^2
]
We finally check that the image is of the expected type.
> P6<x,y,z,s,t,u,v> := ProjectiveSpace(Rationals(),6);
> div_map := map<S->P6|div_map_eqns>;
> X := Image(div_map);
> Xeqns := MinimalBasis(Ideal(X));
> Xeqns;
[
t^2 - s*u,
z*t - y*u,
z*s - y*t,
z^2 - x*u,
y*z - x*t,
y^2 - x*s,
x^2*z + 2*x^2*t - 2*x*y*t + 2*y*s*t + x*z*u + 2*x*t*u +
y*t*u - s*t*u - y*u^2 + 2*z*u^2 - 2*s*u^2 - u^3 +
2*s*t*v - 2*t*u*v - z*v^2,
x^2*y + 2*x^2*s - 2*x*y*s + 2*y*s^2 + x*y*u + 2*x*s*u +
y*s*u - s^2*u - y*t*u - 2*s*t*u + 2*y*u^2 - t*u^2 +
2*s^2*v - 2*s*u*v - y*v^2,
x^3 - 6*x^2*s + 4*x*y*s + 2*x*s^2 - 4*y*s^2 + x^2*u -
3*x*s*u - 3*y*s*u + 2*s^2*u - x*t*u + 4*s*t*u +
2*x*u^2 - 4*y*u^2 - z*u^2 + 2*t*u^2 + 2*y*s*v -
4*s^2*v - 2*y*u*v + 4*s*u*v - x*v^2 + 2*y*v^2
]
As expected, there are 3 cubics. Also, the quadrics do not contain v so define a singular cone over the scheme they define in the P5 hyperplane v=0. Can see by inspection that this is a Veronese surface. However, we can check this mechanically by the characterisation that it is a non-singular, irreducible, non-degenerate (not lying in a hyperplane) surface of degree 4 with K2=9. We verify the K2 condition with the general machinery for surfaces with at most simple singularities. We also check directly that X has only an A1 singularity at the cone vertex, [0:0:0:0:0:0:1].
> P5 := ProjectiveSpace(Rationals(),5); > hm := hom<CoordinateRing(P6)->CoordinateRing(P5)| > [P5.i : i in [1..6]] cat [0]>; > Xquad := Surface(P5,[hm(Xeqns[i]) : i in [1..6]]); > // The surface constructor has checked irreducibility > // and dimension! > IsNonsingular(Xquad); true > Degree(Xquad); 4 > ChernNumber(Xquad,1); // K^2 = 9 9 > // Check non-degeneracy : no linear form vanishes on Xquad > Saturate(~Xquad); > Min([TotalDegree(f) : f in Basis(Ideal(Xquad))]); 2 1 > HasOnlySimpleSingularities(X : ReturnList := true); true [* <(0 : 0 : 0 : 0 : 0 : 0 : 1), "A", 1> *]
This example deals with blowing down lines that pass through singular points on a degenerate Del Pezzo surface. On non-singular Del Pezzos, a disjoint collection of lines {Li} can be blown down by applying the divisor map of the divisor class H + ∑i Li, where H is a hyperplane divisor. The divisor map can be obtained from either the sheaf DivisorMap intrinsic or from the Riemann-Roch basis computed using the divisor machinery.
In the singular case, the Riemann-Roch space computation in general produces a space of functions that is too large and which doesn't give the anti-canonical embedding of the blow-down. As in the last example, we can get the correct Riemann-Roch space for the blow-down map using linear system divisor restriction.
Our example is a degree 4 Del Pezzo S in P4 with an A1 and an A2 singularity. We have two lines L1 and L2 that we want to blow down which intersect at the A2 singularity but whose strict transforms are (-1)-curves that do not intersect on the desingularised Del Pezzo. The Riemann-Roch space for H + L1 + L2 is of dimension one too large, but gives the correct anticanonical map into P6 after restriction.
Since we want the strict transform of our divisor on S to be a full divisor of the Cartier hyperplane divisor class added to the strict transforms of the two lines, we have to be careful to take a hyperplane on S that doesn't pass through the singular points. Otherwise, divisor restriction leads to a Riemann-Roch space that is too small: the subspace of the required space of functions that vanish to some positive multiplicity on the (-2)-curves above the singular points.
> P4<x,y,z,t,u> := ProjectiveSpace(Rationals(),4);
> S := Surface(P4,[z^2+x*t-y*t-z*u, x*z-y*z+x*u]);
> HasOnlySimpleSingularities(S : ReturnList := true);
true
[* <(0 : 0 : 0 : 1 : 0), "A", 2>, <(1 : 1 : 0 : 0 : 0), "A", 1> *]
> dsds := ResolveSingularSurface(S);
> L1 := Scheme(P4,[x,y,z]);
> L2 := Scheme(P4,[x-y,z,u]);
> MultiplicitiesAndIntersections(S,L1);
[
[ 0 ],
[ 1, 1 ]
]
[
[ 0 ],
[ 1, 0 ]
]
> MultiplicitiesAndIntersections(S,L2);
[
[ 1 ],
[ 1, 1 ]
]
[
[ 1 ],
[ 0, 1 ]
]
We see from this that for the two (-2)-curves over the A2 singularity,
the strict transform of L1 passes through one (transversally) and
that of L2 passes through the other. The strict transform of L2
passes through the A1 singularity and meets the (-2)-curve above it,
whereas that of L1 doesn't. In particular, the strict transforms of
the two lines don't meet.
We first show that if we take a very bad hyperplane divisor (x=0) that passes through both of the singularities, divisor restriction gives a Riemann-Roch space that is much too small.
> D := Divisor(S,L1)+Divisor(S,L2)+Divisor(S,Scheme(S,x));
> rr_fns := RiemannRochBasis(D);
> #rr_fns; // too large by one
8
> den := LCM([Denominator(f): f in rr_fns]);
> den;
y*z - x*u
> B := [R!(den*f) : f in rr_fns] where R is CoordinateRing(P4);
> ms := Multiplicities(S,Scheme(S,den)); ms;
[
[ 1 ],
[ 3, 2 ]
]
> B1 := LinearSystemDivisorRestriction(S,B,ms);
> #B1; // too small by 5!
2
We now get the correct result for blow-down to a non-degenerate degree 6 Del Pezzo
using a hyperplane divisor that avoids the singular points (x + t=0).
> D := Divisor(S,L1)+Divisor(S,L2)+Divisor(S,Scheme(S,x+t));
> rr_fns := RiemannRochBasis(D);
> #rr_fns; // too large by one again
8
> den := LCM([Denominator(f): f in rr_fns]);
> den;
y*z + z*t - x*u
> B := [R!(den*f) : f in rr_fns] where R is CoordinateRing(P4);
> ms := Multiplicities(S,Scheme(S,den)); ms;
[
[ 1 ],
[ 1, 1 ]
]
> B1 := LinearSystemDivisorRestriction(S,B,ms);
> #B1; // correct!
7
B1 gives the blow-down map to the anti-canonical image in P6.
> P6<a,b,c,d,e,f,g> := ProjectiveSpace(Rationals(),6);
> bdmp := map<S->P6|B1>;
> X := Image(bdmp);
> MinimalBasis(Ideal(X));
[
e*f - d*g,
d*f - f^2 - a*g,
c*f - b*g,
d*e - c*g - f*g,
a*e + a*g - b*g,
d^2 - f^2 - b*g,
c*d - b*e,
a*d + a*f - b*f,
a*c - b*d + b*f
]
> Degree(X);
6
> ChernNumber(X,1); //K^2 for X
6