(Geometric) Genus of a curve equals to -1 in SageMath
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
Not a smooth curve.
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.