algcurves
genus
compute the genus of an algebraic curve
Calling Sequence
Parameters
Description
Examples
genus(f, x, y)
f
-
squarefree polynomial specifying an algebraic curve
x, y
variables
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.
withalgcurves:
f≔x4+x2y+y2
factorf
x4+x2y+y2
genusf,x,y
Warning, negative genus so the curve is reducible
−1
evalaAFactorf
x2+RootOf_Z2+32+12yx2+−RootOf_Z2+32+12y
f≔subsz=1,761328152x6z4−5431439286x2y8+2494x2z8+228715574724x6y4+9127158539954x10−15052058268x6y2z2+3212722859346x8y2−134266087241x8z2−202172841y8z2−34263110700x4y6−6697080y6z4−2042158x4z6−201803238y10+12024807786x4y4z2−128361096x4y2z4+506101284x2z2y6+47970216x2z4y4+660492x2z6y2−z10−474z8y2−84366z6y4:
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.
10
See Also
AIrreduc
algcurves[differentials]
algcurves[parametrization]
algcurves[singularities]
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