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algcurves

  

genus

  

The genus of an algebraic curve

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

genus(f, x, y, opt)

Parameters

f

-

squarefree polynomial specifying an algebraic curve

x, y

-

variables

opt

-

(optional) a sequence of options

Description

• 

The genus of an irreducible algebraic curve is a non-negative integer. It equals the dimension of the holomorphic differentials. It also equals (d-1)(d-2)/2 minus the sum of the delta invariants, which can be computed with algcurves[singularities]. Here d is the degree of the curve.

• 

The polynomial f must be squarefree and have degree at least 1, otherwise an error message follows. A complete irreducibility check is not performed, only a few partial tests.

Examples

withalgcurves:

fx4+x2y+y2

f:=x4+x2y+y2

(1)

factorf

x4+x2y+y2

(2)

genusf,x,y

Warning, negative genus so the curve is reducible

1

(3)

evalaAFactorf

x2+12RootOf_Z2+3+12yx2+12RootOf_Z2+3+12y

(4)

fsubsz=1,761328152x6z45431439286x2y8+2494x2z8+228715574724x6y4+9127158539954x1015052058268x6y2z2+3212722859346x8y2134266087241x8z2202172841y8z234263110700x4y66697080y6z42042158x4z6201803238y10+12024807786x4y4z2128361096x4y2z4+506101284x2z2y6+47970216x2z4y4+660492x2z6y2z10474z8y284366z6y4:

This f is a polynomial of degree 10 having a maximal number of cusps according to the Plucker formulas. It was found by Rob Koelman. It has 26 cusps and no other singularities, hence the genus is (10-1)*(10-2)/2 - 26 = 10.

genusf,x,y

10

(5)

See Also

AIrreduc

algcurves[differentials]

algcurves[parametrization]

algcurves[singularities]

 


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