# Genus of an isolated curve on a surface

Let $$S$$ be a smooth projective surface, and $$C \subset S$$ a smooth curve. Take the field to be $$\mathbb{C}$$, so that the curve is a Riemann surface. Now suppose additionally that $$C$$ admits no deformations inside $$S$$ (so that the normal bundle of $$C$$ has no sections, $$H^0(C,N_{C/S})=0$$, or these deformations are obstructed by $$H^1(C,N_{C/S})$$).

My question is: are there bounds on the genus of the curve $$C$$ for a given surface $$S$$?

In many simple cases, the genus of such a curve is zero. For example, this is true for any del Pezzo surface, and for any toric surface, such as the Hirzebruch surfaces. More generally, I might expect there to be a bound in terms of properties of the surface $$S$$, such as the Chern numbers $$\int_S c_1(T_S)^2$$ and $$\int_S c_2(T_S)$$.

I would be very happy to be directed to a reference for this.

• What kind of properties of $S$ do you want to assume ? You can start with any curve on a surface and blow up enough and take the proper transform of your curve to get the normal bundle condition . Feb 3, 2021 at 16:04
• Thanks a lot for your comment @Mohan. My expectation might be that there is a bound for example in terms of $\int_S c_1(T_S)^2$ and $\int_S c_2(T_S)$, which could perhaps capture the cases you mention. Feb 3, 2021 at 16:16
• @TedShifrin: for curves of positive genus, it's not true in general that $h^0(N_C)=0$ implies that $C \cdot C <0$. The simplest example is to blow up 9 general points on an elliptic curve and take $C$ to be the proper transform. Feb 4, 2021 at 9:00