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On parametrizing exceptional tangent cones to Prym theta divisors. (English) Zbl 1361.14023

Given a genus \(g\) curve \(C\), Riemann singularity theorem associates the multiplicity of the theta divisor \(\Theta\) of the Jacobian \(J(C)\simeq\mathrm{Pic}^{g-1}(C)\) at a given point \([L]\) to the dimension of sections of the line bundle \(L\), namely \(\mathrm{mult}_{[L]}\Theta=h^0(C,L)\). By identifying the tangent space of the Jacobian at the origin with \(\mathbb PH^0(CK)\simeq \mathbb P^{g-1}\), one parametrizes the tangent cone \(\mathbb PC_{L}\Theta\) with the Abel-Jacobi image of the linear subspaces normal to \([L]\) in \(C^{(g-1)}\). This description of the tangent cone is established by A. Andreotti and A. L. Mayer [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 21, 189–238 (1967; Zbl 0222.14024)] and G. Kempf [Ann. Math. (2) 98, 178–185 (1973; Zbl 0275.14023)]. Moreover, the algebraic equations for the tangent cone in determinantal form are determined by Kempf [loc. cit.], G. Kempf [“Topics on Riemann surfaces”, Univ. Nac. Aut. Mexico, 27 p. (1973); Abelian integrals. México: Universidad Nacional Autónoma de México (1983; Zbl 0541.14023)].
Given the above well-known results for the Jacobian varieties, a natural question is to find the analogical descriptions for the theta divisors of Prym varieties. The multiplicity of a Prym theta divisor at some given point is known to S. Casalaina-Martin [Ann. Math. (2) 170, No. 1, 163–204 (2009; Zbl 1180.14029)], while the Pfaffian algebraic equations are given by R. Smith and R. Varley [Contemp. Math. 397, 215–236 (2006; Zbl 1099.14019)], the authors of this paper. The geometric parametrization of the tangent cone to the Prym theta divisor is only partially known. R. Smith and R. Varley [Pac. J. Math. 201, No. 2, 479–509 (2001; Zbl 1065.14037)] give a parametrization of the tangent cone of the Prym theta divisors at a point that is at worst a nonexceptional singularity by a linear family of linear spaces of known constant dimension.
In this paper, the authors study the case where the Prym theta divisor has exceptional singularies. They give a rough classification of such exceptional singularities into three types called “exceptional”, “very exceptional” and “totally exceptional”, in the terminology of this paper. These three types of exceptional singularities are distinguished by the étale double cover of the curve, and are seen to give “characteristically different structures” on their tangent cones to the Prym theta divisor.
As a main example, the authors give an explicit geometric parametrization of the tangent cone at the unique triple point, which is a very exceptional singularity, on the theta divisor of the intermediate Jacobian of a cubic threefold. However, further work is still needed to give a complete answer to the parametrization problem for each of the three types of the exceptional singularities.

MSC:

14H40 Jacobians, Prym varieties
14K12 Subvarieties of abelian varieties
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References:

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