Abstract. A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of Allcock-Carlson-Toledo and Looijenga-Swierstra is also considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves.

This paper proves the following converse to a theorem of Mumford: Let A be a principally polarized abelian variety of dimension five, whose theta divisor has a unique singular point, and suppose that the multiplicity of the singular point is three. Then A is isomorphic as a principally polarized abelian variety to the intermediate Jacobian of a smooth cubic threefold. The method of proof is to analyze the possible singularities of the theta divisor of A, and ultimately to show that A is the Prym variety of a possibly singular plane quintic. Licensed to Univ of Mississippi. Prepared on Thu Jul 2 06:13:49 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf Licensed to Univ of Mississippi. Prepared on Thu Jul 2 06:13:49 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf CUBIC THREEFOLDS AND ABELIAN VARIETIES OF DIMENSION FIVE 297to P at x is contained in the tangent cone to Θ at x. These singularities are called stable and exceptional respectively. However, as pointed out by Smith and Varley [11], it is better to distinguish instead the two cases: (1) the tangent space to P at x is not contained in the tangent cone to Θ at x and (2) the tangent space to P at x is contained in the tangent cone to Θ at x. In the first case, the multiplicity of Ξ at x is exactly one half the multiplicity of Θ at x, whereas in the second case there is only an inequality. One goal of this paper is to develop tools to handle case (2). While we only discuss here some of the applications to cubic threefolds, the techniques of this paper can be developed much further. For example, one can prove directly that the multiplicity of the singular point of the theta divisor of JX is three and use this to give another proof of the Torelli theorem for cubic threefolds. It is possible to analyze the singularities of Prym theta divisors in many other situations. These applications will be discussed elsewhere, in the thesis of the first author.The outline of the paper is as follows. Section 1 is a general discussion of how to calculate the multiplicity of points of Sing Ξ. The basic idea is to use test curves to determine the multiplicity, and to measure the multiplicity of Ξ along such a curve by relating it to the extendability of sections to infinitesimal neighborhoods of a point. In Section 2, we deal with the case of a smooth curve C and use the methods of Section 1 to show that, if C is a smooth curve of genus 6 such that the theta divisor Ξ of P = P ( C, π) has a unique singular point of multiplicity 3, then C is a plane quintic. Roughly, the idea is as follows: if L is the line bundle on C corresponding to the singular point, then L is fixed by the involution τ on C corresponding to the double cover, and hence L = π * M for some line bundle M on C, necessarily a theta characteristic. Moreover, Clifford's theorem implies that h 0 (...

Abstract. S. Kondo has constructed a ball quotient compactification for the moduli space of non-hyperelliptic genus four curves. In this paper, we show that this space essentially coincides with a GIT quotient of the Chow variety of canonically embedded genus four curves. More specifically, we give an explicit description of this GIT quotient, and show that the birational map from this space to Kondo's space is resolved by the blow-up of a single point. This provides a modular interpretation of the points in the boundary of Kondo's space. Connections with the slope nine space in the Hassett-Keel program are also discussed.

Abstract. In this paper, motivated by a problem posed by Barry Mazur, we show that for smooth projective varieties over the rationals, the odd cohomology groups of degree less than or equal to the dimension can be modeled by the cohomology of an abelian variety, provided the geometric coniveau is maximal. This provides an affirmative answer to Mazur's question for all uni-ruled threefolds, for instance. Concerning cohomology in degree three, we show that the image of the Abel-Jacobi map admits a distinguished model over the rationals.