Very ampleness for Theta on the compactified Jacobian

It follows from [14, Section 17, p. 163] that 3Θ is very ample. In the singular case, D’Souza has constructed a natural compactification J̄0 for the Jacobian J0 of a complete, integral curve over an algebraically closed field [5]. The scheme J̄0 parametrizes torsion-free, rank 1 sheaves of Euler characteristic 0 on X . A natural question in this context is whether there is a canonical Cartier divisor on J̄0 extending the notion of the classical Theta divisor. The above question was partially and independently answered in [6] and [19]. In these two works the same canonical line bundle L on J̄0 and the same global section θ of L are defined. For smooth curves, the zero scheme of θ is the classical Theta divisor Θ. In [19] Soucaris shows that the zero scheme of the restriction of θ to the maximum reduced subscheme of J̄0 is a Cartier divisor. Both [6] and [19] show that L is ample. It remains to determine whether the zero scheme of θ on J̄0 is a Cartier divisor in general, and what is the minimum n such that L is very ample. In this article our main concern is with the latter question. We will show that L is very ample for n at least equal to a specified lower bound (Theorem 7.) If X has at most ordinary nodes or cusps as singularities, then our lower bound is 3. Our main tool is to use theta sections θE associated to vector bundles E on X . The theta sections were used by Faltings [9] to construct the moduli of semistable vector bundles on a smooth, complete curve without using Geometric Invariant Theory (see also [18].) In a forthcoming work [7], [8] we will apply such method to construct the compactified Jacobian for families of reduced curves. The importance of Theorem 7 is that we obtain a canonical projective embedding of J̄0 in P(H (J̄0,L )), for n minimum such that L is very ample. By studying the structure of the homogeneous coordinate ring of J̄0 in P(H (J̄0,L )), maybe in a way analogous to Mumford’s in [15] and [16], we might be able to understand better the algebraic structure of J̄0.