(Geometric) Genus of a curve equals to -1 in SageMath

asked 2019-12-08 05:57:00 -0500

azerbajdzan gravatar image

updated 2019-12-08 06:07:09 -0500

What does it mean when Sage computes genus of a projective plane curve to be -1?

Input:

x,y,z = QQ['x,y,z'].gens()
C = Curve(x^4 + 10*x^2*y*z + 5*y^2*z^2)
C.genus()

Output:

-1
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Comments

Not a smooth curve.

sage: C.is_smooth()
False
FrédéricC gravatar imageFrédéricC ( 2019-12-08 11:47:15 -0500 )edit
1

C is not geometrically irreducible: over $\mathbb{Q}(\sqrt{5})$ it splits into two components. So, the Euler characteristic of the desingularization ends up being $\chi=-4$. Using the formula $\chi=2g-2$, you end up with genus $-1$. I'm not saying this is a mathematically correct answer; just that this is probably how the answer is discovered: basically by computing the degree of the canonical bundle.

nbruin gravatar imagenbruin ( 2019-12-09 02:09:02 -0500 )edit