Cokernel bundles and Fibonacci bundles

We are interested in those bundles $C$ on $\mathbb{P}^N$ which admit a resolution of the form $$0 \to \mathbb{C}^s \otimes E \xrightarrow{\mu} \mathbb{C}^t \otimes F \to C \to 0.$$ In this paper we prove that, under suitable conditions on $(E,F)$, a generic bundle with this form is either simple or canonically decomposable. As applications we provide an easy criterion for the stability of such bundles on $\mathbb{P}^2$ and we prove the stability when $E = \mathcal{O}$, $F = \mathcal{O}(1)$ and $C$ is an exceptional bundle on $\mathbb{P}^N$ for $N \geq 2$.Comment: 23 pages, 1 figure, revised version, to appear in Mathematische Nachrichte

Simplicity of generic Steiner bundles

We prove that a generic Steiner bundle E is simple if and only if the Euler characteristic of the endomorphism bundle of E is less or equal to 1. In particular we show that either E is exceptional or it satisfies the following inequality t\leq(\frac{n+1+\sqrt((n+1)^2-4)}{2})s.Comment: 11 page

Semistability of certain bundles on a quintic Calabi-Yau threefold

In the paper ``Chirality change in string theory'', by Douglas and Zhou, the authors give a list of bundles on a quintic Calabi-Yau threefold. Here we prove the semistability of most of these bundles. This provides examples of string theory compactifications which have a different number of generations and can be connected

On the Alexander-Hirschowitz Theorem

The Alexander-Hirschowitz theorem says that a general collection of $k$ double points in ${\bf P}^n$ imposes independent conditions on homogeneous polynomials of degree $d$ with a well known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a self-contained proof based mainly on previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case $d=3$, where our proof is shorter. We end with an account of the history of the work on this problem.Comment: 29 pages, the proof in the case of cubics has been simplified, three references added, to appear in J. Pure Appl. Algebr

On a notion of speciality of linear systems in P^n

Given a linear system in P^n with assigned multiple general points we compute the cohomology groups of its strict transforms via the blow-up of its linear base locus. This leads us to give a new definition of expected dimension of a linear system, which takes into account the contribution of the linear base locus, and thus to introduce the notion of linear speciality. We investigate such a notion giving sufficient conditions for a linear system to be linearly non-special for arbitrary number of points, and necessary conditions for small numbers of points.Comment: 26 pages. Minor changes, Definition 3.2 slightly extended. Accepted for publication in Transactions of AM

Postulation of general quartuple fat point schemes in P^3

We study the postulation of a general union $Y$ of double, triple, and quartuple points of $\mathbb{P}^3$. We prove that $Y$ has the expected postulation in degree $d\ge 41$, using the Horace differential lemma. We also discuss the cases of low degree with the aid of computer algebra.Comment: 14 pages, to appear in J. Pure App. Algebr