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On the singularities of theta divisors on Jacobians. (English) Zbl 0954.14020

The aim of this article is to study the intersection cohomology \(IH^\bullet(\Theta,{\mathbb Q})\) of the theta divisor \(\Theta\) of the Jacobian of a complex non-hyperelliptic curve \(X\) of genus \(g\). Since the resolution \(S^{g-1}(X)\to \Theta\) given by the Abel-Jacobi map is small, it induces an isomorphism between \(IH^\bullet(\Theta,{\mathbb Q})\) and \(H^\bullet(S^{g-1}(X);{\mathbb Q})\). It is shown that the characteristic variety of the intersection complex (inside the cotangent bundle of \(\text{ Pic}^{g-1}(X)\)) is irreducible. The action of the involution \(L\mapsto \Omega_X^1\otimes L^{-1}\) of \(\Theta\) on \(IH^\bullet(\Theta,{\mathbb Q})\) is completely determined, and is shown not to respect the algebra structure induced by the isomorphism with \(H^\bullet(S^{g-1}(X);{\mathbb Q})\). The proofs use standard bounds on the dimension of the Brill-Noether loci in \(\text{Pic}^{g-1}(X)\).

MSC:

14H40 Jacobians, Prym varieties
14K25 Theta functions and abelian varieties
14H42 Theta functions and curves; Schottky problem
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14H51 Special divisors on curves (gonality, Brill-Noether theory)
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