# canonicalCurve -- Compute a random canonical curve of genus less or equal to 14

## Synopsis

• Usage:
I=(random canonicalCurve)(g,S)
• Inputs:
• g, an integer, the genus
• R, , homogeneous coordinate ring of $\PP^{ g-1}$
• Outputs:
• I, an ideal, of a canonical curve $C$ of genus $g$

## Description

Compute a random canonical curve of genus $g \le{} 14$, based on the proofs of unirationality of $M_g$ by Severi, Sernesi, Chang-Ran and Verra.

 i1 : setRandomSeed "alpha"; i2 : g=14; i3 : FF=ZZ/10007; i4 : R=FF[x_0..x_(g-1)]; i5 : time betti(I=(random canonicalCurve)(g,R)) -- used 5.98479 seconds 0 1 o5 = total: 1 66 0: 1 . 1: . 66 o5 : BettiTally i6 : genus I == g and degree I ==2*g-2 o6 = true

## For the programmer

The object canonicalCurve is an instance of the type RandomObject.