Let X ⊂ P r be a closed scheme in projective space whose homogeneous ideal is generated by quadrics. We say that X (or its ideal I X ) satisfies the condition N 2,p if the syzygies of I X are linear for p steps. We show that if X satisfies N 2,p then a zero-dimensional or one-dimensional intersection of X with a plane of dimension p is 2-regular. This extends a result of Green and Lazarsfeld. We give conditions when the syzygies of X restrict to the syzygies of the intersection. Many of our results also work for ideals generated by forms of higher degree. As applications, we bound the p for which some well-known projective varieties satisfy N 2,p . Another application, carried out by us in a different paper, is a step in the classification of 2-regular reduced projective schemes. Extending a result of Fröberg, we determine which monomial ideals satisfy N 2,p . We also apply Green's 'linear syzygy theorem' to deduce a relation between the resolutions of I X and I X∪Γ for a scheme Γ, and apply the result to bound the number of intersection points of certain pairs of varieties such as rational normal scrolls.

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§0. Introduction.In this paper, we study the equations of projectively embedded abelian surfaces with a polarization of type (1, d). Classical results say that given an ample line bundle L on an abelian surface A, the line bundle L ⊗n is very ample for n ≥ 3, and furthermore, in case n is even and n ≥ 4, the generators of the homogeneous ideal I A of the embedding of A via L ⊗n are all quadratic; a possible choice for a set of generators of I A are the Riemann theta relations.On the other hand, much less is known about embeddings via line bundles L of type (1, d), that is line bundles L which are not powers of another line bundle on A. It is well-known that if d ≥ 5, and A is a general abelian surface, then L is very ample, while L can never be very ample for d < 5. However, even if d ≥ 5, L may not be very ample for special abelian surfaces. We will restrict our attention in what follows only to the general abelian surface and wish to know what form the equations take for such a projectively embedded abelian surface.A few special cases are well-documented in the literature: d = 4, in which case the general surface is a singular octic in P 3 , cf. [BLvS], and d = 5 in which case the abelian surface is described as the zero set of a section of the Horrocks-Mumford bundle [HM], whereas its homogeneous ideal is generated by 3 (Heisenberg invariant) quintics and 15 sextics (cf. Moreover, in case the embedding considered is with level structure of canonical type, we can give a precise symmetric form for these quadrics.Our approach is as follows: given a line bundle L of type (1, d) on the surface A, we consider the product embedding A × A ⊆ P d−1 × P d−1 , with x's as coordinates on the first P d−1 factor and y's as coordinates on the second factor, and construct certain families of matrices M whose entries are bilinear in the variables x 0 , . . . , x d−1 , and y 0 , . . . , y d−1 , and which will drop rank on A × A. Thus setting (y 0 : . . . : y d−1 ) to be some point in A, suitable minors of M will vanish on the surface A. Furthermore, by choosing special values for the parameter (y 0 : . . . : y d−1 ), one can obtain M 's which are anti-symmetric, and hence deduce that suitable pfaffians of M will vanish on A.These matrices prove to be quite ubiquitous: for d even, we produce a family of The matrices are constructed in §2 and some of their properties are also described in the same chapter. In the remaining part of the paper, we discuss the structure of the ideal of abelian surfaces by using degeneration arguments. Thus in §3, we review the most basic facts about degenerations of abelian surfaces and elliptic curves, and in §4, we construct projectively embedded degenerations using Stanley-Reisner ideals. The deepest degenerations of abelian surfaces we make use of are described by Stanley-Reisner ideals coming from certain triangulations of the 2-torus. Combinatorics then help us to understand the ideals of these degenerations. In §5 we study basic facts and determine equations and syzygies for secant varieties o...

Abstract. We give explicit descriptions of the moduli spaces of abelian surfaces with polarizations of type (1, d), for d = 12, 14, 16, 18 and 20. More precisely, in each case we show that a certain choice of moduli space of such abelian surfaces with a partial level structure can be described explicitly and is unirational, and in some cases rational. These moduli spaces with partial level structure are covers of the ordinary moduli spaces, so the Kodaira dimension of the ordinary moduli spaces in these cases is −∞. In addition, we give a few new examples of Calabi-Yau three-folds fibred in abelian surfaces. In the case of d = 20, such Calabi-Yau three-folds play a key role in the description of the abelian surfaces.

The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises and in subjects as diverse as optimization, coding theory, theta functions, and recently in our proof that certain general sets of points fail to Ž satisfy the minimal free resolution conjecture see Eisenbud and Popescu, 1999, . In¨ent. Math. 136, 419᎐449 . In this paper we reexamine the Gale transform in the light of modern algebraic geometry. We give a more general definition in the Ž . context of finite locally Gorenstein subschemes. We put in modern form a number of the more remarkable examples discovered in the past, and we add new constructions and connections to other areas of algebraic geometry. We generalize Goppa's theorem in coding theory and we give new applications to Castelnuovo theory. We also give references to classical and modern sources.