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Edoardo Ballico

Publications and source records attributed to Edoardo Ballico.

At least 163 records · Page 9Linked to original sources
arXiv2010

On the X-rank with respect to linearly normal curves

In this paper we study the $X$-rank of points with respect to smooth linearly normal curves $X\subset \PP n$ of genus $g$ and degree $n+g$. We prove that, for such a curve $X$, under certain circumstances, the $X$-rank of a general point of $X$-border rank equal to $s$ is less or equal than $n+1-s$. In the particular case of $g=2$ we give a complete description of the $X$-rank if $n=3,4$; while if $n\geq 5$ we study the $X$-rank of points belonging to the tangential variety of $X$.

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arXiv2009

Koszul cohomology and singular curves

We investigate Koszul cohomology on irreducible nodal curves. In particular, we prove both Green and Green-Lazarsfeld conjectures for the general k-gonal nodal curve.

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arXiv2009

On Green and Green-Lazarfeld conjectures for simple coverings of algebraic curves

Let X be a smooth genus g curve equipped with a simple morphism f: X -> C, where C is either the projective line or more generally any smooth curve whose gonality is computed by finitely many pencils. Here we apply a method developed by Aprodu to prove that if g is big enough then X satisfies both Green and Green-Lazarsfeld conjectures. We also partially address the case in which the gonality of C is computed by infinitely many pencils.

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arXiv2009

Vector bundles near negative curves: moduli and local Euler characteristic

We study moduli spaces of vector bundles on a two-dimensional neighbourhood $Z_k$ of an irreducible curve $\ell = CP^1$ with $\ell^2 = -k$ and give an explicit construction of these moduli as stratified spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on $Z_k$ and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/~s0571100/Instanton/ .

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arXiv2009

Smoothing of rational m-ropes

In a recent paper, Gallego, González and Purnaprajna showed that rational 3-ropes can be smoothed. We generalise their proof and obtain smoothability of rational $m$-ropes for $m \geq 3$.

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arXiv2008

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.

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arXiv2008

Local moduli of holomorphic bundles

We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi-Yau threefolds.

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arXiv2008

A splitting criterion for vector bundles on blowing ups of the plane

Let $f_s: X_s \to {\bf {P}}^2$ be the blowing-up of $s$ distinct points and $E$ a vector bundle on $X_s$. Here we give a cohomological criterio which is equivalent to $E \cong f_s^\ast (A)$ with $A$ a direct sum of line bundles. We also some cohomological characterizations of very particular rank 2 vector bundles on ${\bf {P}}^2$.

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arXiv2008

Vector Bundles on Products of Varieties with $n$-blocks Collections

Here we consider the product of varieties with $n$-blocks collections . We give some cohomological splitting conditions for rank 2 bundles. A cohomological characterization for vector bundles is also provided. The tools are Beilinson's type spectral sequences generalized by Costa and Miró-Roig. Moreover we introduce a notion of Castelnuovo-Mumford regularity on a product of finitely many projective spaces and smooth quadric hypersurfaces in order to prove two splitting criteria for vector bundle with arbitrary rank.

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arXiv2008

Qregularity and an Extension of Evans-Griffiths Criterion to Vector Bundles on Quadrics

Here we define the concept of Qregularity for coherent sheaves on quadrics. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on $\Q_n\subset \mathbb P^{n+1}$ with the Castelnuovo-Mumford regularity of their extension by zero in $\mathbb P^{n+1}$. We also classify the coherent sheaves with Qregularity $-\infty$. We use our notion of Qregularity in order to prove an extension of Evans-Griffiths criterion to vector bundles on Quadrics. In particular we get a new and simple proof of the Knörrer's characterization of ACM bundles.

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arXiv2007

On the minimal free resolution for fat point schemes of multiplicity at most 3 in P^2

Let Z be a fat point scheme in P^2 supported on general points. Here we prove that if the multiplicities are at most 3 and the length of Z is sufficiently high then the number of generators of the homogeneous ideal I_Z in each degree is as small as numerically possible. Since it is known that Z has maximal Hilbert function, this implies that Z has the expected minimal free resolution.

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arXiv2007

On the birational geometry of moduli spaces of pointed curves

We prove that the moduli space ${\Cal M}_{g,n}$ of smooth curves of genus $g$ with $n$ marked points is rational for $g=6$ and $1 \le n \le 8$, and it is unirational for $g=8$ and $1 \le n \le 11$, $g=10$ and $1 \le n \le 3$, $g=12$ and $n = 1$.

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arXiv2004

On a Lemma of Bompiani

We reprove in modern terms and extend to arbitrary dimension a classical result of Enrico Bompiani on algebraic surfaces $X \subset \PP^r$ having very degenerate osculating spaces.

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arXiv2004

Birational geometry of defective varieties

Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness, rational connectivity, and rationality.

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arXiv2004

Vector bundles on a three dimensional neighborhood of a ruled surface

Let $S$ be a ruled surface inside a smooth threefold $W$ and let $E$ be a vector bundle on a formal neighborhood of $S.$ We find minimal conditions under which the local moduli space of $E$ is finite dimensional and smooth. Moreover, we show that $E$ is a flat limit of a flat family of vector bundles whose general element we describe explicitly.

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