Basepoint-freeness thresholds and higher syzygies of abelian threefolds

Syzygies of Abelian Varieties

Let A be an ample line bundle on an abelian variety X (over an algebraically closed field). A theorem of Koizumi ([Ko], [S]), developing Mumford’s ideas and results ([M1]), states that if m ≥ 3 the line bundle L = A⊗m embeds X in projective space as a projectively normal variety. Moreover, a celebrated theorem of Mumford ([M2]), slightly refined by Kempf ([K4]), asserts that the homogeneous ide...

Vector Bundles and Monads on Abelian Threefolds

Using the Serre construction, we give examples of stable rank 2 vector bundles on principally polarized abelian threefolds (X,Θ) with Picard number 1. The Chern classes (c1, c2) of these examples realize roughly one half of the classes that are a priori allowed by the Bogomolov inequality and Riemann-Roch (the latter gives a certain divisibility condition). In the case of even c1, we study defo...

Local positivity of principally polarized abelian threefolds

Basepoint freeness for nef and big line bundles in positive characteristic

A necessary and sufficient condition is given for semi-ampleness of a numerically effective (nef) and big line bundle in positive characteristic. One application is to the geometry of the universal stable curve over Mg, specifically, the semi-ampleness of the relative dualizing sheaf, in positive characteristic. An example is given which shows this and the semi-ampleness criterion fail in chara...

Higher Syzygies of Elliptic Ruled Surfaces

The purpose of this article is to study the minimal free resolution of homogeneous coordinate rings of elliptic ruled surfaces. Let X be an irreducible projective variety and L a very ample line bundle on X , whose complete linear series defines the morphism φL : X −→ P(H (L)) Let S = ⊕∞ m=0 S H(X,L) and R(L) ⊕∞ m=0 H (X,L). Since R(L) is a finitely generated graded module over S, it has a mini...