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

A lower bound for the number of components of the moduli schemes of stable rank 2 vector bundles on projective 3-folds

Fix a smooth projective 3-fold X, c1, H ∈ Pic(X) with H ample, and d ∈ Z. Assume the existence of integers a, b with a ≠ 0 such that ac1 is numerically equivalent to bH. Let M(X, 2, c1, d, H) be the moduli scheme of H-stable rank 2 vector bundles, E, on X with c1(E) = c1 and c2(E) · H = d. Let m(X, 2, c1, d, H) be the number of its irreducible components. Then lim supd→ ∞m(X, 2, c1, d, H) = +∞

Dimension of families of determinantal schemes

A scheme $X\subset \mathbb{P} ^{n+c}$ of codimension $c$ is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous $t \times (t+c-1)$ matrix and $X$ is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$ we denote by $W(\underline{b};\underline{a})\subset \operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$(resp. $W_s(\underline{b};\underline{a})$) the locus of good (resp. standard) determinantal schemes $X\subset \mathbb{P} ^{n+c}$ of codimension $c$ defined by the maximal minors of a $t\times (t+c-1)$ matrix $(f_{ij})^{i=1,...,t}_{j=0,...,t+c-2}$ where $f_{ij}\in k[x_0,x_1,...,x_{n+c}]$ is a homogeneous polynomial of degree $a_j-b_i$. In this paper we address the following three fundamental problems: To determine (1) the dimension of $W(\underline{b};\underline{a})$ (resp. $W_s(\underline{b};\underline{a})$) in terms of $a_j$ and $b_i$, (2) whether the closure of $W(\underline{b};\underline{a})$ is an irreducible component of $\operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$, and (3) when $\operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$ is generically smooth along $W(\underline{b};\underline{a})$. Concerning question (1) we give an upper bound for the dimension of $W(\underline{b};\underline{a})$ (resp. $W_s(\underline{b};\underline{a})$) which works for all integers $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$, and we conjecture that this bound is sharp. The conjecture is proved for $2\le c\le 5$, and for $c\ge 6$ under some restriction on $a_0,a_1,...,a_{t+c-2}$and $b_1,...,b_t$. For questions (2) and (3) we have an affirmative answer for $2\le c \le 4$ and $n\ge 2$, and for $c\ge 5$ under certain numerical assumptions

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

Togliatti systems and Galois coverings

We study the homogeneous artinian ideals of the polynomial ring generated by the homogeneous polynomials of degree d which are invariant under an action of the cyclic group , for any . We prove that they are all monomial Togliatti systems, and that they are minimal if the action is defined by a diagonal matrix having on the diagonal , where e is a primitive d-th root of the unity. We get a complete description when d is prime or a power of a prime. We also establish the relation of these systems with linear Ceva configurations

The representation type of determinantal varieties

This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves $\mathcal{E}$ of arbitrary high rank on a general standard (resp. linear) determinantal scheme $X \subset \mathbb{P}^n$ of codimension $c \geq 1, n-c \geq 1$ and defined by the maximal minors of a $t \times(t+c-1)$ homogeneous matrix $\mathcal{A}$. The sheaves $\mathcal{E}$ are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme $X \subset \mathbb{P}^n$ is of wild representation type provided the degrees of the entries of the matrix $\mathcal{A}$ satisfy some weak numerical assumptions; and (2) we determine values of $t, n$ and $n-c$ for which a linear standard determinantal scheme $X \subset \mathbb{P}^n$ is of wild representation type with respect to the much more restrictive category of its indecomposable Ulrich sheaves, i.e. $X$ is of Ulrich wild representation type