Commutative ring theory: Regular rings

Basic Algebraic Geometry

Generalized divisors on Gorenstein schemes

We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\bf P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.

Comparing powers and symbolic powers of ideals

The Weak and Strong Lefschetz properties for Artinian K-algebras

Part 1 Background: finitely generated graded S-modules the deficiency modules (Mi)(V) hyperplane and hypersurface sections Artinian reductions and h-vectors examples. Part 2 Submodules of the deficiency module: measuring deficiency generalizing Dubreil's theorem lifting the Cohen-Macaulay property. Part 3 Buchsbaum curves: liaison addition constructing Buchsbaum curves in P3. Part 4 Gorenstein subschemes of Pn: basic results on Gorenstein ideals constructions of Gorenstein schemes - intersection of linked schemes, sections of Buchsbaum-Rim sheaves of odd rank, linear systems on aCM scheme codimension three Gorenstein ideals. Part 5 Liaison theory: definition and first example relations between linked schemes the Hartshorne-Schenzel theorem the structure of an even liaison class geometric invariants of a liaison class. Part 6 Liaison theory in codimension two: the aCM situation and generalizations Rao's results the Lazarsfeld-Rao property applications - smooth curves in P3, smooth surfaces in P4 and threefold in P5, possible degrees and genera in a codimension two even liaison class, stick figures, low rank vector bundles and schemes defined by a small number of equations.

Introduction to Liaison Theory and Deficiency Modules

Introduction Preliminaries Gaeta's theorem Divisors on an ACM subscheme of projective spaces Gorenstein ideals and Gorenstein liaison CI-liaison invariants Geometric applications of the CI-liaison invariants Glicci curves on arithmetically Cohen-Macaulay surfaces Unobstructedness and dimension of families of subschemes Dimension of families of determinantal subschemes Bibliography.

Gorenstein liaison, complete intersection liaison invariants and unobstructedness

Let A = bigoplus_{i >= 0} A_i be a standard graded Artinian K-algebra, where char K = 0. Then A has the Weak Lefschetz property if there is an element ell of degree 1 such that the multiplication times ell : A_i --> A_{i+1} has maximal rank, for every i, and A has the Strong Lefschetz property if times ell^d : A_i --> A_{i+d} has maximal rank for every i and d. 
The main results obtained in this paper are the following. 
1) EVERY height three complete intersection has the Weak Lefschetz property. (Our method, surprisingly, uses rank two vector bundles on P^2 and the Grauert-Mulich theorem.) 
2) We give a complete characterization (including a concrete construction) of the Hilbert functions that can occur for K-algebras with the Weak or Strong Lefschetz property (and the characterization is the same one). 
3) We give a sharp bound on the graded Betti numbers (achieved by our construction) of Artinian K-algebras with the Weak or Strong Lefschetz property and fixed Hilbert function. This bound is again the same for both properties. Some Hilbert functions in fact FORCE the algebra to have the maximal Betti numbers. 
4) EVERY Artinian ideal in K[x,y] possesses the Strong Lefschetz property. This is false in higher codimension.

The Weak and Strong Lefschetz Properties for Artinian K-Algebras

Many algebras are expected to have the Weak Lefschetz property though this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the intriguing dependence of the property on the characteristic of the ground field, and on arithmetic properties of the exponent vectors of the monomials.

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Juan C. Migliore

Papers

Introduction to Liaison Theory and Deficiency Modules

The Weak and Strong Lefschetz properties for Artinian K-algebras

Gorenstein liaison, complete intersection liaison invariants and unobstructedness

The Weak and Strong Lefschetz Properties for Artinian K-Algebras

Monomial ideals, almost complete intersections and the Weak Lefschetz property