We say that an infinite set S of closed points of an integral scheme X is critically dense if every infinite subset of S is Zariski dense in X. This property was introduced by Rogalski, who used it to construct examples of noncommutative algebras with counter-intuitive properties.

Given a closed point p Î X and an automorphism s of X, we look at the set {sⁿ(p) | n Î Z}. Motivated by a question of Rogalski, we explain why for sets of this form, critical density is the same as density for affine n-space over a field of characteristic 0, Fano varieties over a field of characteristic 0, and projective n-space over a field of characteristic 0. Moreover, we explain why these results do not hold in characteristic p > 0. We conclude with some open questions.

I want to consider the k-osculating varieties to the Veronese d-uple embeddings of Pⁿ and study the dimension of their higher secant varieties O^s_k,n,d via Inverse Systems. By associating certain 0-dimensional schemes Y Ì Pⁿ to O^s_k,n,d and by studying their Hilbert function it's possible, in several cases, to determine whether those secant varieties are defective or not.

Free divisors are reduced hypersurfaces whose singular locus is Cohen-Macaulay of codimension 2 in the ambient space. This concept was isolated first by K. Saito in the early seventies, who observed that the classical discriminants of polynomials are of this form and explained this in terms of deformation theory. In this talk, we indicate how new examples arise from the representation theory of quivers.

Parts of this talk are joint work with W. Ebeling (Hanover, Germany) and D. Mond (Warwick, UK).

I will deal with basic module theoretic operations (syzygy computations, module intersection and colon, linear system solving) of free, finitely generated modules over k[x₁,...,x_n]. I will show a technique for reducing these operations to the computation of a suitable component elimination module. The algorithms resulting from this technique are easy to describe, prove and implement without any efficiency loss in comparison with the traditional algorithms.

Let d=2e denote an even positive integer. Identify the space of n-ary d-ics with P^N, where N = \binomd+n-1d-1. Now consider the projective subvariety

The crux of the result turns out to be representation-theoretic, and is reminiscent of the Foulkes-Howe conjecture. It amounts to the assertion that a certain canonical morphism

Given a polynomial ring R = k[x₁,...,x_n], graded by N, and a homogeneous artinian ideal I of R, then the quotient A=R/I is a graded ring that is a finite dimensional vector space over k. In particular, A can be decomposed as A = kÅA₁żÅA_c, where A_c ¹ 0. We let the h-vector of A be the vector (1,h₁,...,h_c), where h_i = dim_k A_i.

It is a natural question to ask which sequences of numbers (1,b₁,...,b_l) can be the h-vector of a graded artinian ring. This question was answered by Macaulay, who gave a test one can apply to the sequence to figure out whether it is the h-vector of a graded artinian ring or not.

However, if one imposes the condition that A must be a level algebra, then question becomes more difficult and it has not been solved yet. In general, provided one knows n, c and h_c, then one can find an upper bound on the size of the h_i. However, there is no corresponding lower bound.

In this talk, we shall examine this issue and compute a lower bound for the dimension of A_c-1. We shall then show that while this lower bound is not sharp, we can get very close to it. In particular, we will show that Zanello's theorem on the size of h_c-1 does not extend to the case n > 7.

A characterization of which sequences can be the Hilbert function of a finite set of distinct points in projective n-space Pⁿ (called valid Hilbert functions for Pⁿ) follows from the work of Macaulay, Hartshorne, and others. Although Hilbert functions of complete intersections are well-known, Hilbert functions of subsets of complete intersections have not yet been classified, even for the simplest cases. Let 1 £ d₁ £ d₂ £ ¼ £ d_n be positive integers and H be a valid Hilbert function for Pⁿ. We wish to determine if there exists some reduced zero-dimensional complete intersection C.I.(d₁, ..., d_n) which contains a subset whose Hilbert function is H.

The special case of this problem where the ideal of the complete intersection is generated by products of linear forms follows from the combinatorial work of Clements and Lindström. In general, this problem was completely answered for the case n=2 in my M.Sc. thesis and I currently have partial results for n=3. We will discuss these results as well as consider the complications that have arisen in working on this research problem. We will conclude with applications to subsets with the Cayley-Bacharach Property and extremal subsets similar to Sub_d(H). This work will be included in my Ph.D. dissertation.

Let R be an integral domain, K its field of fractions and X,Y indeterminates. An R-endomorphism R[X,Y] ® R[X,Y] is said to be generically automorphic if its extension K[X,Y] ® K[X,Y] is an automorphism. Let \operatornameEnd^*_R (R[X,Y]) be the set of R-endomorphisms of R[X,Y] which are generically automorphic. For each r Î R\{0}, define m^(r) Î \operatornameEnd^*_R (R[X,Y]) by m^(r)(X) = X and m^(r)(Y) = rY.

We say that R has the weak GA-property if every element of \operatornameEnd^*_R (R[X,Y]) is a composition of R-automorphisms of R[X,Y] and of endomorphisms of type m^(r). We partially solve the problem of identifying the class of domains having the weak GA-property. In particular, we show that an affine domain over C has the weak GA-property if and only if it is a principal ideal domain.

Joint work with Stéphane Vénéreau.

If R = k[x₀,x₁] = Å_{d ³ 0} R_d is the usual graded decomposition of the polynomial ring, then the map f_(d1,d2): P¹ ×P¹ ® P^{d₁ d₂+d₁+d₂} induced by the canonical multilinear map

The question considered in this talk is: What are the dimensions of all the higher secant varieties to this embedding of P¹ ×P¹?

We show how this question is related to that of finding the Hilbert function, in degree d₁ + d₂, of the subscheme of P² defined by an ideal of the form

We completely solve this problem by giving a description of the entire Hilbert function of schemes defined by ideals like I above. As a byproduct of our investigations we verify the Harbourne-Hirschowitz conjecture for this family of ideals.

This is joint work with M. V. Catalisano (Genoa) and A. Gimigliano (Bologna).

For a commutative local ring R, consider (possibly noncommutative) R-algebras L of the form L = End_R(M), where M is a reflexive R-module with nonzero free direct summand. Such algebras L of finite global dimension can be viewed as possible subsitutes for, or analogues of, a resolution of singularities of SpecR. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal C-algebra with terminal singularities has a crepant resolution of singularities if and only if it has such an algebra L with finite global dimension and which is maximal Cohen-Macaulay over R (a "noncommutative crepant resolution of singularities"). We produce algebras L = End_R(M) having finite global dimension in two contexts: when R is a complete one-dimensional local ring, or when R is a Cohen-Macaulay ring of finite Cohen-Macaulay type. If, in the latter case, R is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh.

In the early 1970s, Hochster proved that normal semigroup rings generated by monomials are Cohen-Macaulay. When we weaken normal to seminormal, naturally, a question is raised.

Question: When is a seminormal affine semigroup ring Cohen-Macaulay? Conversely, when is a Cohen-Macaulay affine semigroup ring seminormal?

However, this question is not easily answered because for affine semigroup rings, many examples show that Cohen-Macaulay and seminormal may not coincide. For example, A=k[t²,t³] is Cohen-Macaulay but not seminormal, conversely, A=k[x,y,z²,xz,yz] is seminormal but not Cohen-Macaulay.

In this talk I will explore the coincidence between the Cohen-Macaulay property and seminormality of arbitrary affine semigroup rings, and demonstrate that under certain circumstances, an affine semigroup ring is seminormal if it is Cohen-Macaulay. Furthermore, a conjecture is given for a seminormal affine semigroup ring k[S] to be Cohen-Macaulay, and is proved when rank(S) \leqslant 3.

This talk will report on work, joint with Craig Huneke and Paul Monsky, regarding the existence of a second coefficient in the Hilbert-Kunz function of an m-primary ideal in a normal local ring of positive characteristic. In particular, let (R, m, k) be an excellent, local, normal ring of characteristic p with a perfect residue field and dimR=d. We let n be a varying non-negative integer and let q=pⁿ. If I is an m-primary ideal of R, and M is a finitely generated R-module, then there exists a real number b such that the length of M/I^[q]M can be written as aq^d +bq^d-1 +O(q^d-2).

I will speak on joint work concerning ring homomorphisms R ® S that provide S with an R-module structure that has an extremal resolution. If S is not finitely generated, we have to generalize the definition of Betti numbers in an appropriate way. (More generally, we consider and develop the homological algebra of an S-module over the homomorphism rather than the special case N=S.) In particular, we get an example of a surprising phenomenon: that the R-module structure of S can actually give information about the rings R and S.

The ur-example of an extremal homomorphism is the Frobenius homomorphism in characteristic p > 0, but there are many others; in fact, the original motivation of this work was to show the extremality of the Frobenius.

Let R be a commutative Noetherian Nagata ring, let M be a non-zero finitely generated R-module, and let I be an ideal of R such that \operatornameheight_M I > 0. In this paper there is a definition of the integral closure N_a for any submodule N of M extending Rees' definition for the case of a domain. As a main result it is shown that for any submodule N of M, the sequences \operatornameAss_R M/(Iⁿ N)_a and \operatornameAss_R (Iⁿ M)_a / (IⁿN)_a, n = 1,2,..., of associated prime ideals, are increasing and ultimately constant for large n.

This talk will describe joint work with Peter Schenzel.

We compare several ways of measuring how far a projective monomial curve is from being Cohen-Macaulay, and use these results to show that as the degree d goes to infinity the number of Cohen-Macaulay projective monomial curves of degree d grows exponentially, but the fraction of all projective monomial curves of degree d that are Cohen-Macaulay approaches 0.

This is joint work with Les Reid.

It seems somewhat unbelievable that there would be a fact about the invariant theory of finite groups in the nonmodular case which is not well understood, but I think that there is an essential principle missing. The principle is simply the invariant theory version of Brauer's theory of relative invariants over a DVR or Dedekind domain.

The natural language to express this principle is that of algebraic geometry, and the talk will focus on explaining what the conjectured principle is, as well as some evidence for its truth, and some of its consequences.

Let R be a commutative Noetherian ring, and let x = x₁,...,x_d be a regular R-sequence contained in the Jacobson radical of R. An ideal I of R is said to be a monomial ideal with respect to x if it is generated by a set of monomials x₁^e₁¼x_d^e_d. The monomial closure of I, denoted by [(I)\tilde], is defined to be the ideal generated by the set of all monomials m such that mⁿ Î Iⁿ for some n Î N. It is shown that the sequences \operatornameAss_R R/[(Iⁿ)\tilde] and \operatornameAss_R[(Iⁿ)\tilde]/Iⁿ, n = 1,2,..., of associated prime ideals are increasing and ultimately constant for large n. In addition, some results about the monomial ideals and their integral closures are included.

We'll discuss how to construct projective simplicial toric varieties as the fine moduli space of representations of a quiver.