What is the polarization type of the push-forward of the Poincaré-bundle to the Jacobian of a curve?

Question

$\DeclareMathOperator{\Jac}{Jac}\DeclareMathOperator{\Pic}{Pic}$Let $C$ be a smooth curve of genus $g > 0$, and consider the Picard torus $\Pic^d(C)$ of line bundles of degree $d$. Let $\mathcal P$ be a Poincaré bundle on $C \times \Pic^d(C)$. If $d > 2g - 2$, every line bundle $L \in \Pic^d(C)$ has $H^1(C, L) = 0$. If we denote by $q: C \times \Pic^d(C) \to \Pic^d(C)$ the projection, it follows that $$R^1q_* \mathcal P = 0,$$ and $q_* \mathcal P$ is a locally free sheaf of rank $r = \dim H^0(C,L) = d - g + 1$.

I would like to know the polarization type $(d_1, \dotsc, d_g)$ of $$c_1(q_* \mathcal P) \in H^2(\mathbb Z, \Pic^d(C)).$$ Does anyone know how to compute this, or if there is any reference for this?

Answer 1

$\DeclareMathOperator{\Pic}{Pic}\DeclareMathOperator{\Sym}{Sym}$I found an answer in a paper by Schwarzenberger[1].

Fix a point $c \in C$, and for $n \in \mathbb Z$ consider on $\Pic^0(C)$ the sheaves $$E_n = q_*(\mathcal P^0 \otimes p^*\mathcal O(nc)) \quad \text{and} \quad F_n = R^1q_*(\mathcal P^0 \otimes p^*\mathcal O(nc)).$$ Schwarzenberger proves for $n > 2g-2$ $$\mathbb P(E_n^\vee) \cong \Sym^n(C),$$ and the map $\mathbb P(E_n^\vee) \to \Pic^0(C)$ coincides with the composition of $\rho: \Sym^n(C) \to \Pic^n(C)$ with $\Pic^n(C) \to \Pic^0(C), L \mapsto L \otimes \mathcal O(-nc)$. By work of Mattuck[2] (which is also reproduced in [1]), this implies $$c_1(E_n) = - [W_{g-1}],$$ where $W_{g-1}$ is the image of $\Sym^{g-1}(C)$ in $\Pic^0(C)$. By Poincaré's formula [3, 11.2.1], the class $[W_{g-1}]$ is homologic to the class of a Theta divisor. Hence $c_1(E_n)$ gives minus a principal polarization on $\Pic^0(C)$.

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[1] R. L. E. Schwarzenberger, Jacobians and symmetric products, Illinois J. Math. 7 (1963), 257--268; MR0151459,

[2] A. Mattuck, Symmetric products and Jacobians, Amer. J. Math. 83 (1961), 189--206; MR0142553

[3] C. Birkenhake and H. Lange, Complex abelian varieties, second edition, Grundlehren der mathematischen Wissenschaften, 302, Springer, Berlin, 2004; MR2062673