Let I be a homogeneous ideal of S = K[x 1 , . . . , x n ] and let < be a term order. We prove that if the initial ideal J = in < (I) is radical then the extremal Betti numbers of S/I and of S/J coincide. In particular, depth(S/I) = depth(S/J) and reg(S/I) = reg(S/J).

ABSTRACT. It is well-known that the Plücker relations generate the ideal of relations of the maximal minors of a generic m × n matrix. In this paper we discuss the relations of t-minors for t < min(m, n). We will exhibit minimal relations in degrees 2 (non-Plücker in general) and 3, and give some evidence for our conjecture that we have found the generating system of the ideal of relations. The approach is through the representation theory of the general linear group.

ABSTRACT. An ideal I in a polynomial ring S has linear powers if all the powers I k of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that every rational normal scroll has linear powers.

Abstract. In this paper we partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a long-standing conjecture of Stanley. As a byproduct of this theory we establish Stanley's conjecture in various cases, for example the case of Cohen-Macaulay type less than or equal to 3.

Given an equidimensional algebraic set X ⊂ P n , its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne's connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshorne's result:(1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo-Mumford regularity of X. (The bound is best possible. For coordinate arrangements, it yields an algebraic extension of Balinski's theorem for simplicial polytopes.) (2) If X is an arrangement of lines no three of which meet in the same point, and X is canonically embedded in P n , then the diameter of the graph G(X) is less than or equal to codim P n X.(The bound is sharp; for coordinate arrangements, it yields an algebraic expansion on the recent combinatorial result that the Hirsch conjecture holds for flag normal simplicial complexes.) On the way to these results, we show that there exists a graph which is not the dual graph of any simplicial complex (no matter the dimension).The dual graph need not be connected, as shown for example by the ideal I = (x, y) ∩ (z, w) inside C[x, y, z, w], whose dual graph consists of two disjoint vertices. The reader familiar with combinatorics should note that this ideal is monomial and squarefree, so via the Stanley-Reisner correspondence it can be viewed as a simplicial complex. There is already an established notion of "dual graph of a (pure) simplicial complex" and it is compatible with our definition, in the sense that if I ∆ is the Stanley-Reisner ideal of a pure complex ∆, the dual graphs of ∆ and of I ∆ are the same. This way it is usually easy to produce examples of ideals with prescribed dual graphs. However, not all graphs are dual graphs of a simplicial complex, as we will see in Corollary 4.2.Having connected dual graph is a property well studied in the literature under the name of "connectedness in codimension one". Remarkably, it is shared by all Cohen-Macaulay algebras: Theorem 1.1 (Hartshorne [Har62]). For any ideal I ⊂ S, if S/I is Cohen-Macaulay then G(I) is connected.(It is well known that, if S/I is Cohen-Macaulay, then I is height-unmixed). But can we say more about how connected G(I) is, if we know more about I -for example, that I is generated in certain degrees, or that S/I is Gorenstein? This leads to the following question. Problem 1.2. Give a quantitative version of Hartshorne's connectedness theorem.There are at least two natural directions to explore: (a) lower bounds for the connectivity, and (b) upper bounds for the diameter.Connectivity counts how many vertex-disjoint paths there are (at least) between two arbitrary points of the graph. Balinski's theorem says that the graph of every d-polytope is d-connected. Since the dual graph of any d-polytope P is also the 1-skeleton of a d-polytope (namely, of the polar polytope P * ), an equivalent reformulation of Balinski's theor...