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Suppose $C$ is a genus $4$ smooth projective curve over complex numbers. The Abel-Jacobi map from $Sym^4(C)$ to $Jac(C)$ is birational. Is this a blow-up along a surface or a curve. Can one determine this surface or curve?

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The locus in $Jac(C) = Pic^4(C)$ over which the map is not an isomorphism is the locus of degree 4 line bundles with $h^0 \ge 2$. By Serre duality this is the same as the locus of degree 2 line bundles with $h^0 \ge 2$, i.e., the image of $Sym^2(C)$ under the map $$ Sym^2(C) \to Pic^4(C), \qquad (P,Q) \mapsto K_C - P - Q.\tag{$*$} $$ If $C$ is not hyperelliptic, this map is a closed embedding and thus $$ Sym^4(C) \cong Bl_{Sym^2(C)} Jac(C). $$ If $C$ is hyperelliptic, $Sym^2(C)$ contains a smooth rational curve (the hyperelliptic linear system) that is contracted by the map $(*)$, and the fiber of the Abel-Jacobi map over the corresponding point of $Jac(C)$ is $\mathbb{P}^2$. Most probably, the Abel-Jacobi map is still the blowup of the singular surface, the image of $Sym^2(C)$ under $(*)$.

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  • $\begingroup$ many thanks for the explanation $\endgroup$
    – john
    Jun 21 '19 at 9:23
  • $\begingroup$ @Sasha can you provide some reference for this implication "closed embedding => blow up"?. I can't figure out why this map is actually a blow up and not some other map satisfying given properties (e.g. weighted blow up)?. $\endgroup$ Sep 11 '20 at 22:06
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In any genus, it is true that the Abel-Jacobi map $\operatorname{Sym}^g(C)\rightarrow \operatorname{Jac}^g(C)$ is the blow up of the integral subvariety $W^1_g$ of $\operatorname{Jac}^g(C)$ parametrizing line bundles $L$ with $h^0(L)\geq 2$. This is a result of Muñoz-Porras, On the structure of the birational Abel morphism, Math. Ann. 281 (1988), no. 1, 1-6.

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  • $\begingroup$ many thanks for the explanation and reference. $\endgroup$
    – john
    Jun 21 '19 at 9:22

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