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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.

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    $\begingroup$ 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 . $\endgroup$
    – Mohan
    Feb 3, 2021 at 16:04
  • $\begingroup$ 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. $\endgroup$
    – diracula
    Feb 3, 2021 at 16:16
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    $\begingroup$ @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. $\endgroup$ Feb 4, 2021 at 9:00

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