An n×n nilpotent matrix B is determined up to conjugacy by a partition PB of n, its Jordan type given by the sizes of its Jordan blocks. The Jordan type D(P) of a nilpotent matrix in the dense orbit of the nilpotent commutator of a given nilpotent matrix of Jordan type P is stable - has parts differing pairwise by at least two - and was determined by R. Basili. The second two authors, with B. Van Steirteghem and R. Zhao determined a rectangular table of partitions D−1(Q) having a given stable partition Q as the Jordan type of its maximum nilpotent commutator. They proposed a box conjecture, that would generalize the answer to stable partitions Q having ℓ parts: it was proven recently by J. Irving, T. Košir and M. Mastnak. Using this result and also some tropical calculations, the authors here determine equations defining the loci of each partition in D−1(Q), when Q is stable with two parts. The equations for each locus form a complete intersection. The authors propose a conjecture generalizing their result to arbitrary stable Q.

The rate of a standard graded K-algebra A is a measure of the growth of the shifts in a minimal free resolution of K as an A-module. In particular A has rate one if and only if it is Koszul. It is known that a generic Artinian Gorenstein algebra of embedding dimension n≥3 and socle degree s=3 is Koszul. We prove that a generic Artinian Gorenstein algebra with n≥4 and s≥3 has rate s2. In the process we show that such an algebra is generated in degree s2+1. This gives a partial positive answer to a longstanding conjecture stated by the first author on the minimal free resolution of a generic Artinian Gorenstein ring of odd socle degree

We use Macaulay's inverse system to study the Hilbert series for almost complete intersections generated by uniform powers of general linear forms. This allows us to give a classification of the weak Lefschetz property for these algebras, settling a conjecture by Migliore, Miro-Roig, and Nagel.

We consider the conjecture that all artinian height 4 complete intersections of forms of the same degree d have the Weak Lefschetz Property (WLP). We translate this problem to one of studying the general hyperplane section of a certain smooth curve in P3, and our main tools are the Socle Lemma of Huneke and Ulrich together with a careful liaison argument. Our main results are (i) a proof that the property holds for d = 3, 4 and 5; (ii) a partial result showing maximal rank in a non-trivial but incomplete range, cutting in half the previous unknown range; and (iii) a proof that maximal rank holds in a different range, even without assuming that all the generators have the same degree. We furthermore conjecture that if there were to exist any height 4 complete intersection generated by forms of the same degree and failing WLP then there must exist one (not necessarily the same one) failing by exactly one (in a sense that we make precise). Based on this conjecture we outline an approach to proving WLP for all equigenerated complete intersections in four variables. Finally, we apply our results to the Jacobian ideal of a smooth surface in P3.

We show that the complete symmetric polynomials are dual generators of compressed artinian Gorenstein algebras satisfying the strong Lefschetz property. This is the first example of an explicit dual form with these properties. For complete symmetric forms of any degree in any number of variables, we provide an upper bound for the Waring rank by establishing an explicit power sum decomposition. Moreover, we determine the Waring rank, the cactus rank, the resolution and the strong Lefschetz property for any Gorenstein algebra defined by a symmetric cubic form. In particular, we show that the difference between the Waring rank and the cactus rank of a symmetric cubic form can be made arbitrarily large by increasing the number of variables. We provide upper bounds for the Waring rank of generic symmetric forms of degrees four and five.

We show that the Waring rank of the 3 x 3 determinant, previously known to be between 14 and 18, is at least 15. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the symmetric cactus rank of the 3 x 3 permanent is at least 14.

We determine a sharp lower bound for the Hilbert function in degree d of a monomial algebra failing the weak Lefschetz property over a polynomial ring with n variables and generated in degree d, for any d≥2 and n≥3. We consider artinian ideals in the polynomial ring with n variables generated by homogeneous polynomials of degree d invariant under an action of the cyclic group Z/dZ, for any n≥3 and any d≥2. We give a complete classification of such ideals in terms of the weak Lefschetz property depending on the action.

Mustata has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in P-3 this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.

Macaulay's theorem and Fröberg's conjecture deal with the Hilbert function of homogeneous ideals in polynomial rings over a field K. In this short note we present some questions related to variants of Macaulay's theorem and Fröberg's conjecture for K-subalgebras of polynomial rings.

Given an ideal I=(f1,…,fr) in C[x1,…,xn] generated by forms of degree d, and an integer k>1, how large can the ideal Ik be, i.e., how small can the Hilbert function of C[x1,…,xn]/Ik be? If r≤n the smallest Hilbert function is achieved by any complete intersection, but for r>n, the question is in general very hard to answer. We study the problem for r=n+1, where the result is known for k=1. We also study a closely related problem, the Weak Lefschetz property, for S/Ik, where I is the ideal generated by the d'th powers of the variables.