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Let $S$ be a connected smooth projective surface over $\mathbb{C}$.

Let $\Sigma$ be the complete linear system of a very ample divisor $D$ on $S$, and let $d=\dim(\Sigma)$ be the dimension of $\Sigma$. Recall that $\Sigma$ can be identified with $(\mathbb{P}^d)^*$, the dual of the projective space $\mathbb{P}^d$, which parametrizes hyperplanes in $\mathbb{P}^d$.

For each close point $t\in\Sigma$, let $H_t$ be the corresponding hyperplane in $\mathbb{P}^d$ and let $C_t=S\cap H_t$ the corresponding hyperplane section of $S$. (Note that since $S$ is a surface, $C_t$ is a curve on $S$).

I would like to know what is the behaviour of the genus $g(C_t)$ of the smooth curves $C_t$ (on $S$) which are members of the linear system $\Sigma$. I mean, Do all the members $C_t$ that are smooth have the same genus? or these members can have different genus?

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    $\begingroup$ All smooth sections have the same genus $g$, given by the adjunction formula: $g=1+\frac{1}{2}(D^2+D\cdot K_S) $. $\endgroup$
    – abx
    Commented Aug 8, 2022 at 12:35
  • $\begingroup$ Dear @abx Thank you for your answer! So, this implies that the Jacobians $J_t$ of all the smooth curves $C_t$ have the same dimension, right? However, by the definition of the Jacobian of a curve, the Jacobians are not necessarily isomorphic, right? $\endgroup$
    – Roxana
    Commented Aug 8, 2022 at 18:42
  • $\begingroup$ Yes and yes.... $\endgroup$
    – abx
    Commented Aug 8, 2022 at 19:09
  • $\begingroup$ Dear @abx thank you for your answer! Another question: I think that the curve $C_{\overline{\eta}}$ corresponding to the geometric generic point $\overline{\eta}$ of $(\mathbb{P}^d)^*$ is always assumed to be smooth. If I am correct, I would like to know why $C_{\overline{\eta}}$ is smooht? $\endgroup$
    – Roxana
    Commented Aug 9, 2022 at 14:23
  • $\begingroup$ Look for Bertini theorem. $\endgroup$
    – abx
    Commented Aug 9, 2022 at 16:25

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