Subcanonical Curves

A subcanonical curve is a polarised variety C, D where C is a nonsingular curve of genus g≥2 and D is a divisor on C such that KC = kD for some positive integer k.

Contents

Creation of Subcanonical Curves

SubcanonicalCurve(g,d,Q) : RngIntElt,RngIntElt,SeqEnum -> GRCrvK
The subcanonical curve C, D of genus g, degree d and initial Hilbert series coefficients Q.
IsSubcanonicalCurve(g,d,Q) : RngIntElt,RngIntElt,SeqEnum -> BoolElt,GRCrvK
Return true if and only if the data g, d, Q passes some basic checks that there is a subcanonical curve C, D of genus g, degree d and initial Hilbert series coefficients Q. In that case, the second return value is such a curve.
HilbertPolynomialOfCurve(g,m) : RngIntElt,RngIntElt -> RngUPolElt
The Hilbert polynomial mt + 1 - g of a divisor of degree m on a curve of genus g.
IsEffective(C) : GRCrvK -> BoolElt
Return true if and only if the polarising divisor of the subcanonical curve C is effective; that is, if and only if the Hilbert series has the form 1 + p1t + ... with p1>0.

Catalogue of Subcanonical Curves

This section describes intrinsics that allow the user to generate many examples of Hilbert series of subcanonical curves and attempt to interpret them as curves embedded in wps.

EffectiveSubcanonicalCurves(g) : RngIntElt -> SeqEnum
EffectiveSubcanonicalCurves(g,d) : RngIntElt,RngIntElt -> SeqEnum
A sequence containing data for effective subcanonical curves of genus g≥3 (polarised by a divisor of degree d if the second argument is given).
IneffectiveSubcanonicalCurves(g) : RngIntElt -> SeqEnum
IneffectiveSubcanonicalCurves(g,d) : RngIntElt,RngIntElt -> SeqEnum
A sequence containing data for ineffective subcanonical curves of genus g≥3 (polarised by a divisor of degree d if the second argument is given).
V2.28, 13 July 2023