Cubic threefolds and abelian varieties of dimension five

Abstract

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 (...

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

“…a characterization of the intermediate Jacobian locus I, was obtained by Friedman and the first author in [20]: the intermediate Jacobians of cubic threefolds are precisely those five dimensional ppavs whose theta divisor has a unique singular point, which is of multiplicity three. It is then shown in [19] that the condition of a triple point on the theta divisor cuts in A 5 three ten dimensional irreducible components, one of which isĪ ∩ A 5 .…”

Singularities of the Prym theta divisor

“…In this section we state the key results from [CMF05], which we will need in what follows. The proofs of these facts will be omitted except in the cases where certain generalizations are needed.…”

“…In [CMF05], a converse to Mumford's theorem was proved: if .A; ‚/ is a ppav of dimension 5, Sing ‚ D fxg and mult x " D 3, then .A; ‚/ is isomorphic to the intermediate Jacobian of a smooth cubic threefold. If one removes the condition that Sing ‚ D fxg and requires instead the weaker condition that exactly one of the singular points of ‚ has multiplicity 3, then it was shown that the only other possibility is that .A; ‚/ is isomorphic to J C or J C J C 0 for some hyperelliptic curves C and C 0 (i.e., curves having a line bundle L such that deg.…”

Cubic threefolds and abelian varieties of dimension five. II

“…The following was proven in [6]. Lemma 1.1.1 Suppose S is a smooth curve, f : S → P is a morphism, and f (s 0 ) = x ∈ P corresponds to a line bundle L ∈ P * .…”

“…Section 1 recalls the basic setup in [5,6]; a more general situation is considered in this paper, and the technical difficulties this introduces are dealt with in Sect. 1 …”