Bressler, P.; Brylinski, J.-L. On the singularities of theta divisors on Jacobians. (English) Zbl 0954.14020 J. Algebr. Geom. 7, No. 4, 781-796 (1998). 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 \(\text{Pic}^{g-1}(X)\). Reviewer: O.V.Debarre (Strasbourg) Cited in 4 Documents 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) Keywords:Jacobians; intersection cohomology; characteristic variety; theta divisor; Abel-Jacobi map; Brill-Noether loci; Picard bundle PDFBibTeX XMLCite \textit{P. Bressler} and \textit{J. L. Brylinski}, J. Algebr. Geom. 7, No. 4, 781--796 (1998; Zbl 0954.14020) Full Text: arXiv