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I'm trying to construct complete smooth curves $C$ in $\overline{M}_g$ such that for all points $S \in C \cap \mathcal{M}_g$, its Jacobian $\text{Jac}(S)$ satisfies $\text{End}(\text{Jac}(S)) = \mathbb{Z}$. Does anyone know a source of these?

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    $\begingroup$ Welcome new contributor. You used the letter $C$ to denote the base curve in the compactified moduli space $\overline{\mathcal{M}}_g$. Then you used the same letter for the curves whose Jacobians you study. Did you intend to write that the Jacobians of the (varying) fiber curves should all have "trivial" endomorphism algebra? $\endgroup$
    – Jason Starr
    Commented Aug 10, 2023 at 11:31
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    $\begingroup$ @JasonStarr I think Gina meant to ask about Jac(S) instead of Jac(C). I went ahead and corrected the body of the question. $\endgroup$
    – Ariyan Javanpeykar
    Commented Aug 10, 2023 at 12:11
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    $\begingroup$ @JasonStarr: Your interpretation is correct. Sorry for the typo! I read it multiple times before posting, but somehow the eye always manages to fill in what it wants to see rather than what is there... $\endgroup$
    – Gina
    Commented Aug 10, 2023 at 15:29
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    $\begingroup$ @AriyanJavanpeykar: Thanks for correcting that! $\endgroup$
    – Gina
    Commented Aug 10, 2023 at 15:29
  • $\begingroup$ I have a related question. Is it true that for a general point p = [C] in M_g, we have that the endomorphism ring of Jac(C) is Z? $\endgroup$
    – Cranium Clamp
    Commented Aug 13, 2023 at 3:21

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