Uniform Steiner bundles

In this work we study $k$-type uniform Steiner bundles, being $k$ the lowest degree of the splitting. We prove sharp upper and lower bounds for the rank in the case $k=1$ and moreover we give families of examples for every allowed possible rank and explain which relation exists between the families. After dealing with the case $k$ in general, we conjecture that every $k$-type uniform Steiner bundle is obtained through the proposed construction technique.Comment: 23 pages, to appear at Annales de l'Institut Fourie

Families of determinantal schemes

Given integers a_0 \le a_1 \le ... \le a_{t+c-2} and b_1 \le ... \le b_t, we denote by W(b;a) \subset Hilb^p(\PP^{n}) the locus of good determinantal schemes X \subset \PP^{n} of codimension c defined by the maximal minors of a t x (t+c-1) homogeneous matrix with entries homogeneous polynomials of degree a_j-b_i. The goal of this short note is to extend and complete the results given by the authors in [10] and determine under weakened numerical assumptions the dimension of W(b;a), as well as whether the closure of W(b;a) is a generically smooth irreducible component of the Hilbert scheme Hilb^p(\PP^{n}).Comment: The non-emptiness of W(b;a) is restated as (2.2) in this version; the codimension c=2 case in (2.5)-(2.6) is reconsidered, and c > 2 (c > 3) is now an assumption in (2.16)-(2.17). 13 page

Stability of syzygy bundles

We show that given integers $N$, $d$ and $n$ such that ${N\ge2}$, ${(N,d,n)\ne(2,2,5)}$, and ${N+1\le n\le\tbinom{d+N}{N}}$, there is a family of $n$ monomials in $K[X_0,\ldots,X_N]$ of degree $d$ such that their syzygy bundle is stable. Case ${N\ge3}$ was obtained independently by Coand\v{a} with a different choice of families of monomials [Coa09]. For ${(N,d,n)=(2,2,5)}$, there are $5$ monomials of degree~$2$ in $K[X_0,X_1,X_2]$ such that their syzygy bundle is semistable.Comment: 16 pages, to appear in the Proceedings of the American Mathematical Societ

Minimal Resolution of Relatively Compressed Level Algebras

A relatively compressed algebra with given socle degrees is an Artinian quotient $A$ of a given graded algebra R/\fc, whose Hilbert function is maximal among such quotients with the given socle degrees. For us \fc is usually a ``general'' complete intersection and we usually require that $A$ be level. The precise value of the Hilbert function of a relatively compressed algebra is open, and we show that finding this value is equivalent to the Fr\"oberg Conjecture. We then turn to the minimal free resolution of a level algebra relatively compressed with respect to a general complete intersection. When the algebra is Gorenstein of even socle degree we give the precise resolution. When it is of odd socle degree we give good bounds on the graded Betti numbers. We also relate this case to the Minimal Resolution Conjecture of Mustata for points on a projective variety. Finding the graded Betti numbers is essentially equivalent to determining to what extent there can be redundant summands (i.e. ``ghost terms'') in the minimal free resolution, i.e. when copies of the same $R(-t)$ can occur in two consecutive free modules. This is easy to arrange using Koszul syzygies; we show that it can also occur in more surprising situations that are not Koszul. Using the equivalence to the Fr\"oberg Conjecture, we show that in a polynomial ring where that conjecture holds (e.g. in three variables), the possible non-Koszul ghost terms are extremely limited. Finally, we use the connection to the Fr\"oberg Conjecture, as well as the calculation of the minimal free resolution for relatively compressed Gorenstein algebras, to find the minimal free resolution of general Artinian almost complete intersections in many new cases. This greatly extends previous work of the first two authors.Comment: 31 page