The underlying field is $\mathbb{C}$. Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\mathcal{L}$ be an invertible sheaf on $\mathcal{C}$ of relative degree $0$ (i.e., for all $t \in \mathbb{A}^n$, $\deg(\mathcal{L}_t)=0$, where $\mathcal{L}_t$ is the restriction of $\mathcal{L}$ to $\mathcal{C}_t$). Denote by $T$ the subscheme of $\mathbb{A}^n$ parametrizing all points $t \in \mathbb{A}^n$ such that $h^0(\mathcal{L}_t) \not=0$. Is $T$ regular? If not true in general is there any condition on $\mathcal{C}$ under which this holds true? Any reference/hint will be most welcome.
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$\begingroup$ Does it matter for you whether the base of your family of curves is $\mathbb{A}^n$ or not? $\endgroup$– SashaCommented May 11, 2018 at 14:07
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$\begingroup$ @Sasha Not much. $\endgroup$– user43198Commented May 11, 2018 at 14:12
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$\begingroup$ Are you willing to make a base change? You can take a blow-up of $\mathbb A^n$ that resolves $T$ and then pull everything back along it; maybe this is enough for what you're trying to do. I don't know if what you want is true in general, but I think there are pretty easy counterexamples when the fibers are 2-dimensional instead of curves. $\endgroup$– user47305Commented May 11, 2018 at 14:33
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$\begingroup$ @Mark Ok, I am trying to avoid resolutions. I am more interested in general criterion one could use to check regularity of $T$. $\endgroup$– user43198Commented May 11, 2018 at 14:39
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1 Answer
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For a general smooth base $S$ it is not true that $T$ is regular. Assume even that $\mathcal{C} = C \times S$ for a fixed smooth curve $C$. The line bundle $\mathcal{L}$ then defines a morphism $f \colon S \to Pic^0(C)$ and then $T = f^{-1}(\{0\})$. Conversely, given a morphism $f$ recovers back the line bundle $\mathcal{L}$. Thus, in case of $\mathcal{C} = C \times S$, the scheme $T$ is just the preimage of the origin in $Pic^0(C)$ under an arbitrary map $f \colon S \to Pic^0(C)$.
To construct an example when $T = f^{-1}(\{0\})$ is singular, one can consider the blowup of $Pic^0(C) \times \mathbb{A}^1$ in a point over $\{0\}$. In this case $T$ is even reducible.
