arXiv subjects

David Eisenbud

Publications and source records attributed to David Eisenbud.

60 records · Page 4Linked to original sources

arXiv1998

The Projective Geometry of the Gale Transform

The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises, in subjects as diverse as optimization, coding theory, theta-functions, and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture. In this paper we reexamine the Gale transform in the light of modern algebraic geometry. We give a more general definition, in the 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 give references to classical and modern sources.

math.AG↗

arXiv1997

Gale Duality and Free Resolutions of Ideals of Points

What is the shape of the free resolution of the ideal of a general set of points in P^r? This question is central to the programme of connecting the geometry of point sets in projective space with the structure of the free resolutions of their ideals. There is a lower bound for the resolution computable from the (known) Hilbert function, and it seemed natural to conjecture that this lower bound would be achieved. This is the ``Minimal Resolution Conjecture'' (Lorenzini [1987], [1993]). Hirschowitz and Simpson [1994] showed that the conjecture holds when the number of points is large compared with r, but three examples (with r = 6,7,8) discovered computationally by Schreyer in 1993 show that the conjecture fails in general. We describe a novel structure inside the free resolution of a set of points which accounts for the observed failures and provides a counterexample in P^r for every r\geq 6, r\neq 9. The geometry behind our construction occurs not in P^r but in a different projective space, in which there is a related set of points, the ``Gale transform'' (or ``associated set'', in the sense of Coble.)

alg-geom↗

arXiv1995

Evolutions, Symbolic Squares, and Fitting Ideals

Given a reduced local algebra $T$ over a suitable ring or field $k$ we study the question of whether there are nontrivial algebra surjections $R\to T$ which induce isomorphisms $Ω_{R/k}\otimes T \to Ω_{T/k}$. Such maps, called evolutions, arise naturally in the study of Hecke algebras, as they implicitly do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the existence of non-trivial evolutions of an algebra $T$ can be characterized in terms of the symbolic square of an ideal defining $T$. We give a characterization of the symbolic square in terms of Fitting ideals. Using this and other techniques we show that certain classes of reduced algebras -- codimension 2 Cohen-Macaulay, Codimension 3 Gorenstein, licci algebras in general, and some others -- admit no nontrivial evolutions. On the other hand we give examples showing that non-trivial evolutions of reduced Cohen-Macaulay algebras of codimension 3 do exist in every positive characteristic.

alg-geom↗

arXiv1994

Binomial Ideals

We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and it has numerous applications within and beyond pure mathematics. The ideals defining toric varieties are precisely the binomial prime ideals. Our main results concern primary decomposition: If $I$ is a binomial ideal then the radical, associated primes, and isolated primary components of $I$ are again binomial, and $I$ admits primary decompositions in terms of binomial primary ideals. A geometric characterization is given for the affine algebraic sets that can be defined by binomials. Our structural results yield sparsity-preserving algorithms for finding the radical and primary decomposition of a binomial ideal.

alg-geom↗

arXiv1993

Initial ideals, Veronese subrings, and rates of algebras

We show that high Veronese subrings of any commutative graded ring have a Grobner basis with all relations of degree 2. (The d-th Veronese subring of a ring A_0 + A_1 + A_2 + ... is the ring A_0 + A_d + A_{2d} + ...; ``high'' means we take d sufficiently large, say at least half the regularity of the ideal defining the original ring.) This gives another proof of Backelin's theorem that such Veronese subrings are Koszul algebras (= wonderful rings), i.e., that the minimal resolution of the residue field of such a ring is linear.

alg-geom↗

arXiv1993

Finding Sparse Systems of Parameters

For several computational procedures such as finding radicals and Noether normalizations, it is important to choose as sparse as possible a system of parameters in a polynomial ideal or modulo a polynomial ideal. We describe new strategies for these tasks, thus providing solutions to problems (1) and (2) posed in [Eisenbud-Huneke-Vasconcelos 1992].

alg-geom↗