arXiv subjects
Publications and source records attributed to Edoardo Ballico.
Let $\mathcal M_{r,c}$ denote the moduli space of stable bundles with rank $r$ and second Chern class $c>0$ on a Hopf surface. We prove that the subset of $\mathcal M_{r,c}$ formed by irregular bundles is meager.
The Hadamard rank of a point with respect to a projective variety is, if it exists, the minimum number of points of the variety whose coordinate-wise product is the given point. We classify the projective varieties for which the Hadamard rank is finite for any point. As a by-product we obtain the finiteness of the Hadamard rank with respect to varieties of tensors, such as Grassmannians, Chow varieties, varieties of reducible forms and their secant varieties, complementing previous known results on secant varieties of Segre-Veronese varieties. We prove sharp upper bounds on the maximum Hadamard rank for certain families of algebraic varieties: this is a consequence of a result on the lower semi-continuity of the Hadamard rank for curves that do not contain points with at least two zero coordinates.
We study projective surfaces in $\mathbb{P}^3$ which can be written as Hadamard product of two curves. We show that quadratic surfaces which are Hadamard product of two lines are smooth and tangent to all coordinate planes, and such tangency points uniquely identify the quadric. The variety of such quadratic surfaces corresponds to the Zariski closure of the space of symmetric matrices whose inverse has null diagonal. For higher-degree surfaces which are Hadamard product of a line and a curve we show that the intersection with the coordinate planes is always non-transversal.
We introduce and study the base locus and the strong base locus of a projective variety X. The base locus of X parametrizes configurations of smooth points of X where the span of the tangent spaces of X at these points intersects X at some additional smooth point. The strong base locus parametrizes configurations of smooth points of X for which the span of the tangent spaces of X at the given configuration contains the entire tangent space at an additional point. These notions originate from the study of base loci of tangential projections, are strictly related to interpolation problems with double points in special position, and provide a natural framework to study tangential contact for nongeneral points. We give first properties and explore connections with Terracini loci and with the concept of identifiability. We focus on tensor-related varieties and characterize the nonemptiness of base loci and strong base loci for Veronese and Segre-Veronese varieties.
We prove a generic flatness result for the cohomology of thickenings of a projective scheme that is smooth over a Noetherian domain containing a field of characteristic zero. Our study is motivated, in part, by a classical question in algebraic geometry: Given a set of $m$ distinct points in projective space over a field, and $t$ a positive integer, determine the least degree of a hypersurface that passes through each point with multiplicity at least $t$. Related to this, it remains unresolved whether there exists a dense open set of $m$-tuples of points for which this least degree is constant for each $t\ge 1$. Investigating this connection in the case of nine points in projective plane, we construct a local cohomology module that is not generically free; moreover, we show that it has infinitely many associated prime ideals.
We show that all filtrable bundles on a Hopf surface $X$ must have jumps and we prove the existence of filtrable stable bundles on $X$ with any value of $c_2>0$. On a somewhat opposite direction, for each integer $r\ge 2$ we prove the existence of irreducible rank $r$ vector bundles on $X$ with trivial determinant, $c_2=1$, and no jumps. We then apply elementary operations in codimension $2$ to points of the moduli space $\mathcal M_{r,n}$ of rank $r$ stable vector bundles on $X$ with $c_2=n$ to obtain torsion free sheaves with $c_2=n+1$. Namely, starting with a surjection $v\colon E \rightarrow \mathbb C_p$ from a vector bundle $E \in \mathcal M_{r,n}$ to a skyscraper sheaf supported at a point $p\in X$, we prove that if $E'$ is any torsion free sheaf fitting into a short exact sequence of the form $0 \longrightarrow E'\longrightarrow E\stackrel{v}{\longrightarrow}\mathbb C_p \longrightarrow 0,$ then $E'$ is in the closure of $\mathcal M_{r,n+1}$. We discuss various properties of vector bundles and torsion free sheaves and introduce the concept of very irreducible bundles to describe bundles whose symmetric powers $S^n(E)$ are irreducible for all $n> 0$. We then show that any rank $2$ bundle on $X$ whose graph contains a component corresponding to a surjective morphism $\mathbb P^1\to \mathbb P^1$ is very irreducible.
We describe the geometry of noncommutative deformations of local Calabi-Yau threefolds, showing that the choice of Poisson structure strongly influences the geometry of the quantum moduli space.
We discuss the hypersurfaces of the moduli spaces of rank $2$ vector bundles on a classical Hopf surface formed by irregular bundles.
The Terracini locus $\mathbb{T}(n, d; x)$ is the locus of all finite subsets $S$ of $ \mathbb{P}^n$ of cardinality $x$ such that $\langle S \rangle = \mathbb{P}^n$, $h^0(\mathcal{I}_{2S}(d)) > 0$, and $h^1(\mathcal{I}_{2S}(d)) > 0$. The celebrated Alexander-Hirschowitz Theorem classifies the triples $(n,d,x)$ for which $\dim\mathbb{T}(n, d; x)=xn$. Here we fully characterize the next step in the case $n=2$, namely, we prove that $\mathbb{T}(2,d;x)$ has at least one irreducible component of dimension $2x-1$ if and only if either $(d,x)\in\{(4,4),(4,6),$ $(5,6),(5,7),$ $(6,9),(6,10)\}$, or $d\ge 7$, $d\equiv 1,2 \pmod{3}$ and $x=(d+2)(d+1)/6$.
Denoting $\mathcal{H}_{d,g,5}$ by the Hilbert scheme of smooth curves of degree $d$ and genus $g$ in $\mathbb{P}^5$, let $\mathcal{H}$ be an irreducible component of $\mathcal{H}_{d,g,5}$. We study the Hilbert function $h_X:\mathbb{N}\longrightarrow\mathbb{N}$, $h_X(t):= h^0(\mathcal{I}_X(t))$ of a general member $X\in\mathcal{H}$, especially when the degree of the curve is low; $d\le 15$. We also determine the irreducibility of $\mathcal{H}_{d,g,5}$ for $d\le 14$ and study the natural functorial map $μ:$\mathcal{H}_{d,g,5}$ \longrightarrow \mathcal{M}_g$ in some detail. We describe the fibre $μ^{-1}μ(X)$ for a general $X\in\mathcal{H} $ as well as determining the projective normality (or being ACM).
We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth, irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r.$ In this article, we study $\mathcal{H}_{15,g,5}$ for every possible genus $g$ and determine when it is irreducible. We also study the moduli map $\mathcal{H}_{15,g,5}\rightarrow\mathcal{M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements of each component.
We study the dimensions of Hadamard products of $k\ge 3$ varieties if we allow to modify $k-1$ of them by the action of a general projective linear transformation. We also prove that the join of a variety not contained in a coordinate hyperplane with a ``nice'' curve always has the expected dimension.
We study minimally Terracini finite sets of points in the projective plane and we prove that the sequence of the cardinalities of minimally Terracini sets can have any number of gaps for degree great enough.
We characterize the number of points for which there exist non-empty Terracini sets of points in $\mathbb{P}^n$. Then we study minimally Terracini finite sets of points in $\mathbb{P}^n$ and we obtain a complete description in the case of $\mathbb{P}^3$, when the number of points is less than twice the degree of the linear system.
Using elementary methods of algebraic geometry, we present constructions of hyperelliptically fibred surfaces containing nodal fibres.
Starting from an integral projective variety $Y$ equipped with a very ample, non-special and not-secant defective line bundle $\mathcal{L}$, the paper establishes, under certain conditions, the regularity of $(Y \times \mathbb P^2,\mathcal{L}[t])$ for $t\geq 2$. The mildness of those conditions allow to classify all secant defective cases of any product of $(\mathbb P^1)^{ j}\times (\mathbb P^2)^{k}$, $j,k \geq 0$, embedded in multidegree at least $(2, \ldots , 2)$ and $(\mathbb{P}^m\times\mathbb{P}^n\times (\mathbb{P}^2)^k, \mathcal{O}_{\mathbb{P}^m\times\mathbb{P}^n\times (\mathbb{P}^2)^k} (d,e,t_1, \ldots, t_k))$ where $d,e \geq 3$, $t_i\geq 2$, for any $n$ and $m$.
We describe deformations of vector bundles on surfaces that are a product of two smooth projective curves. We explicitly describe the Kuranishi map around unstable vector bundles and compare the homologies of the Kuranishi spaces of stable and unstable deformations.
We study surfaces of bidegree (1,d) contained in the flag threefold in relation to the twistor projection. In particular, we focus on the number and the arrangement of twistor fibers contained in such surfaces. First, we prove that there is no irreducible surface of bidegree (1,d) containing d+2 twistor fibers in general position. On the other hand, given any collection of (d+1) twistor fibers satisfying a mild natural constraint, we prove the existence of a surface of bidegree (1,d) that contains them. We improve our results for d=2 or d=3, by removing all the generality hypotheses.