arXiv subjects

Alex Massarenti

Publications and source records attributed to Alex Massarenti.

54 records · Page 3Linked to original sources

arXiv2017

Spherical blow-ups of Grassmannians and Mori Dream Spaces

In this paper we classify weak Fano varieties that can be obtained by blowing-up general points in prime Fano varieties. We also classify spherical blow-ups of Grassmannians in general points, and we compute their effective cone. These blow-ups are, in particular, Mori dream spaces. Furthermore, we compute the stable base locus decomposition of the blow-up of a Grassmannian in one point, and we show how it is determined by linear systems of hyperplanes containing the osculating spaces of the Grassmannian at the blown-up point, and by the rational normal curves in the Grassmannian passing through the blown-up point.

math.AG↗

arXiv2017

On the biregular geometry of the Fulton-MacPherson compactification

Let $X[n]$ be the Fulton-MacPherson compactification of the configuration space of $n$ ordered points on a smooth projective variety $X$. We prove that if either $n\neq 2$ or $\dim(X)\geq 2$, then the connected component of the identity of $Aut(X[n])$ is isomorphic to the connected component of the identity of $Aut(X)$. When $X = C$ is a curve of genus $g(C)\neq 1$ we classify the dominant morphisms $C[n]\rightarrow C[r]$, and thanks to this we manage to compute the whole automorphism group of $C[n]$, namely $Aut(C[n])\cong S_n\times Aut(C)$ for any $n\neq 2$, while $Aut(C[2])\cong S_2\ltimes (Aut(C)\times Aut(C))$. Furthermore, we extend these results on the automorphisms to the case where $X = C_1\times ... \times C_r$ is a product of curves of genus $g(C_i)\geq 2$. Finally, using the techniques developed to deal with Fulton-MacPherson spaces, we study the automorphism groups of some Kontsevich moduli spaces $\overline{M}_{0,n}(\mathbb{P}^N,d)$.

math.AG↗

arXiv2017

Varieties of sums of powers and moduli spaces of (1,7)-polarized abelian surfaces

We study the geometry of some varieties of sums of powers related to the Klein quartic. This allows us to describe the birational geometry of certain moduli spaces of abelian surfaces. In particular we show that the moduli space $\mathcal{A}_2(1,7)^{-}_{sym}$ of $(1,7)$-polarized abelian surfaces with a symmetric theta structure and an odd theta characteristic is unirational by showing that it admits a dominant morphism from a unirational conic bundle.

math.AG↗

arXiv2017

Effective identifiability criteria for tensors and polynomials

A tensor $T$, in a given tensor space, is said to be $h$-identifiable if it admits a unique decomposition as a sum of $h$ rank one tensors. A criterion for $h$-identifiability is called effective if it is satisfied in a dense, open subset of the set of rank $h$ tensors. In this paper we give effective $h$-identifiability criteria for a large class of tensors. We then improve these criteria for some symmetric tensors. For instance, this allows us to give a complete set of effective identifiability criteria for ternary quintic polynomial. Finally, we implement our identifiability algorithms in Macaulay2.

math.AG↗

arXiv2017

Codimension one Fano distributions on Fano manifolds

In this paper we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. As a consequence, we obtain a classification of codimension one del Pezzo distributions on Fano manifolds with Picard number one.

math.AG↗

arXiv2017

On the rigidity of moduli of weighted pointed stable curves

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the Hassett moduli stack of weighted stable curves, and let $\overline{M}_{g,A[n]}$ be its coarse moduli space. These are compactifications of $\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively, obtained by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \leq 1$ to the markings; they are defined over $\mathbb{Z}$, and therefore over any field. We study the first order infinitesimal deformations of $\overline{\mathcal{M}}_{g,A[n]}$ and $\overline{M}_{g,A[n]}$. In particular, we show that $\overline{M}_{0,A[n]}$ is rigid over any field, if $g\geq 1$ then $\overline{\mathcal{M}}_{g,A[n]}$ is rigid over any field of characteristic zero, and if $g+n > 4$ then the coarse moduli space $\overline{M}_{g,A[n]}$ is rigid over an algebraically closed field of characteristic zero. Finally, we take into account a degeneration of Hassett spaces parametrizing rational curves obtained by allowing the weights to have sum equal to two. In particular, we consider such a Hassett $3$-fold which is isomorphic to the Segre cubic hypersurface in $\mathbb{P}^4$, and we prove that its family of first order infinitesimal deformations is non-singular of dimension ten, and the general deformation is smooth.

math.AG↗

arXiv2016

On non-secant defectivity of Segre-Veronese varieties

Let $SV^{\pmb n}_{\pmb d}$ be the Segre-Veronese given as the image of the embedding induced by the line bundle $\mathcal{O}_{\mathbb{P}^{n_1}\times\dots\times\mathbb{P}^{n_r}}(d_1,\dots, d_r)$. We prove that asymptotically $SV^{\pmb n}_{\pmb d}$ is not $h$-defective for $h\leq n_1^{\lfloor \log_2(d-1)\rfloor}$.

math.AG↗

arXiv2016

Non-Secant Defectivity via Osculating Projections

We introduce a method to produce bounds for the non secant defectivity of an arbitrary irreducible projective variety, once we know how its osculating spaces behave in families and when the linear projections from them are generically finite. Then we analyze the relative dimension of osculating projections of Grassmannians, and as an application of our techniques we prove that asymptotically the Grassmannian $\mathbb{G}(r,n)$, parametrizing $r$-planes in $\mathbb{P}^n$, is not $h$-defective for $h\leq (\frac{n+1}{r+1})^{\lfloor\log_2(r)\rfloor}$. This bound improves the previous one $h\leq \frac{n-r}{3}+1$, due to H. Abo, G. Ottaviani and C. Peterson, for any $r\geq 4$.

math.AG↗

arXiv2016

Moduli of abelian surfaces, symmetric theta structures and theta characteristics

We study the birational geometry of some moduli spaces of abelian varieties with extra structure: in particular, with a symmetric theta structure and an odd theta characteristic. For a $(d_1,d_2)$-polarized abelian surface, we show how the parities of the $d_i$ influence the relation between canonical level structures and symmetric theta structures. For certain values of $d_1$ and $d_2$, a theta characteristic is needed in order to define Theta-null maps. We use these Theta-null maps and preceding work of other authors on the representations of the Heisenberg group to study the birational geometry and the Kodaira dimension of these moduli spaces.

math.AG↗

arXiv2015

On the automorphisms of Hassett's moduli spaces

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the moduli stack parametrizing weighted stable curves, and let $\overline{M}_{g,A[n]}$ be its coarse moduli space. These spaces have been introduced by B. Hassett, as compactifications of $\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively, by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \leq 1$ to the markings. In particular, the classical Deligne-Mumford compactification arises for $a_1 = ... = a_n = 1$. In genus zero some of these spaces appear as intermediate steps of the blow-up construction of $\overline{M}_{0,n}$ developed by M. Kapranov, while in higher genus they may be related to the LMMP on $\overline{M}_{g,n}$. We compute the automorphism groups of most of the Hassett's spaces appearing in the Kapranov's blow-up construction. Furthermore, if $g\geq 1$ we compute the automorphism groups of all Hassett's spaces. In particular, we prove that if $g\geq 1$ and $2g-2+n\geq 3$ then the automorphism groups of both $\overline{\mathcal{M}}_{g,A[n]}$ and $\overline{M}_{g,A[n]}$ are isomorphic to a subgroup of $S_{n}$ whose elements are permutations preserving the weight data in a suitable sense.

math.AG↗

arXiv2015

On the rigidity of moduli of curves in arbitrary characteristic

The stack $\overline{\mathcal{M}}_{g,n}$ of stable curves and its coarse moduli space $\overline{M}_{g,n}$ are defined over $\mathbb{Z}$, and therefore over any field. Over an algebraically closed field of characteristic zero, Hacking showed that $\overline{\mathcal{M}}_{g,n}$ is rigid (a conjecture of Kapranov). Bruno and Mella for $g=0$, and the second author for $g\geq 1$ showed that its automorphism group is the symmetric group $S_n$, permuting marked points unless $(g,n)\in\{(0,4),(1,1),(1,2)\}$. The methods used in the papers above do not extend to positive characteristic. We show that in characteristic $p>0$, the rigidity of $\overline{\mathcal{M}}_{g,n}$, with the same exceptions as over $\mathbb{C}$, implies that its automorphism group is $S_n$. We prove that, over any perfect field, $\overline{M}_{0,n}$ is rigid and deduce that, over any field, $Aut(\overline{M}_{0,n})\cong S_{n}$ for $n\geq 5$. Going back to characteristic zero, we prove that for $g+n>4$, the coarse moduli space $\overline M_{g,n}$ is rigid, extending a result of Hacking who had proven it has no locally trivial deformations. Finally, we show that $\overline{M}_{1,2}$ is not rigid, although it does not admit locally trivial deformations, by explicitly computing his Kuranishi family.

math.AG↗

arXiv2015

Explicit log Fano structures on blow-ups of projective spaces

In this paper we determine which blow-ups $X$ of $\mathbb{P}^n$ at general points are log Fano, that is, when there exists an effective $\mathbb{Q}$-divisor $Δ$ such that $-(K_X+Δ)$ is ample and the pair $(X,Δ)$ is klt. For these blow-ups, we produce explicit boundary divisors $Δ$ making $X$ log Fano.

math.AG↗

arXiv2014

Generalized varieties of sums of powers

Let $X\subset\mathbb{P}^{N}$ be an irreducible, non-degenerate variety. The generalized variety of sums of powers $VSP_H^X(h)$ of $X$ is the closure in the Hilbert scheme $Hilb_{h}(X)$ of the locus parametrizing collections of points $\{x_{1},...,x_{h}\}$ such that the $(h-1)$-plane $\left\langle x_{1},...,x_{h}\right\rangle$ passes trough a fixed general point $p\in\mathbb{P}^{N}$. When $X = V_{d}^{n}$ is a Veronese variety we recover the classical variety of sums of powers $VSP(F,h)$ parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of $VSP_H^X(h)$. In particular we will show how some birational properties, such as rationality, unirationality and rational connectedness, of $VSP_H^X(h)$ are inherited from the birational geometry of variety $X$ itself.

math.AG↗

arXiv2013

On the rank of $n\times n$ matrix multiplication

For every $p\leq n$ positive integer we obtain the lower bound $(3-\frac{1}{p+1})n^2-\big(2\binom{2p}{p+1}-\binom{2p-2}{p-1}+2\big)n$ for the rank of the $n\times n$ matrix multiplication. This bound improves the previous one $(3-\frac{1}{p+1})n^2-\big(1+2p\binom{2p}{p}\big)n$ due to Landsberg. Furthermore our bound improves the classic bound $\frac{5}{2}n^2-3n$, due to Bläser, for every $n\geq 132$. Finally, for $p = 2$, with a sligtly different strategy we menage to obtain the lower bound $\frac{8}{3}n^2-7n$ which improves Bläser's bound for any $n\geq 24$.

cs.CC↗

arXiv2013

On the automorphisms of moduli spaces of curves

In the last years the biregular automorphisms of the Deligne-Mumford's and Hassett's compactifications of the moduli space of n-pointed genus g smooth curves have been extensively studied by A. Bruno and the authors. In this paper we give a survey of these recent results and extend our techniques to some moduli spaces appearing as intermediate steps of the Kapranov's and Keel's realizations of $\bar{M}_{0,n}$, and to the degenerations of Hassett's spaces obtained by allowing zero weights.

math.AG↗

arXiv2011

Covered by Lines and Conic Connected Varieties

We study some properties of an embedded variety covered by lines and give a numerical criterion ensuring the existence of a singular conic through two of its general points. We show that our criterion is sharp. Conic-connected, covered by lines, QEL, LQEL, prime Fano, defective, and dual defective varieties are closely related. We study some relations between the above mentioned classes of objects using celebrated results by Ein and Zak.

math.AG↗

arXiv2011

Varieties Connected by Chains of Lines

In this paper we give for any integer l > 2 a numerical criterion ensuring the existence of a chain of length l of lines through two general points of an irreducible variety X in P^N, involving the degrees and the number of homogeneous polynomials defining X. We show that our criterion is sharp.

math.AG↗

arXiv2010

Birational aspects of the Geometry of Varieties of Sum of Powers

Varieties of Sums of Powers describe the additive decompositions of an homogeneous polynomial into powers of linear forms. Despite their long history, going back to Sylvester and Hilbert, few of them are known for special degrees and number of variables. In this paper we aim to understand a general birational behaviour of VSP, if any. To do this we birationally embed these varieties into Grassmannians and prove the rationality, unirationality or rational connectedness of many of those in arbitrary degrees and number of variables.

math.AG↗