geometricGenus -- Geometric genus of a (singular) plane curve

Synopsis

Usage:

g = geometricGenus R

g = geometricGenus I
Inputs:
- R, a ring, homogeneous coordinate ring of a plane curve
- I, an ideal, defining a plane curve
- Conductor, an ideal, the conductor of R (if not given, it's computed)
Optional inputs:
- Conductor => ..., default value 0
Outputs:
- g, an integer,

Description

The geometric genus of a plane curve C0 is the genus of the normalization of C0

i1 : kk = QQ

o1 = QQ

o1 : Ring

i2 : S = kk[x,y,z]

o2 = S

o2 : PolynomialRing

i3 : C1 = ideal (y^3 - x^2*(x-z)) -- cubic with a node; geometric genus 0

              3    3    2
o3 = ideal(- x  + y  + x z)

o3 : Ideal of S

i4 : C2 = ideal(x^2+y^2+z^2)

            2    2    2
o4 = ideal(x  + y  + z )

o4 : Ideal of S

i5 : C3 = ideal (x^4+y^4+z^4)

            4    4    4
o5 = ideal(x  + y  + z )

o5 : Ideal of S

i6 : geometricGenus C1

o6 = 0

i7 : geometricGenus C2

o7 = 0

i8 : geometricGenus C3

o8 = 3

Every hyperelliptic curve of genus g can be represented as a plane curve of degree g+2 with a g-fold ordinary singularity, and thus conductor equal to the (g-1)st power of the maximal ideal. As of 1/20/2024, Macaulay2 crashes on computing the conductor when g >= 6, but knowing the conductor one can go much farther:

We make a general hyperelliptic curve of genus g with singularity at q'.

Example g = 20 S = ZZ/101[a,b,c] q' = ideal(a,b); Text Example I = q'^g C = S/(ideal random(g+2, I)); p = sub(p', C); q = sub(q', C); geometricGenus (C, Conductor => q'^(g-1))

Ways to use geometricGenus :

geometricGenus(Ideal)
geometricGenus(Ring)

For the programmer

The object geometricGenus is a method function with options.

geometricGenus -- Geometric genus of a (singular) plane curve

Synopsis

Description

See also

Ways to use geometricGenus :

For the programmer