Papers by Massimiliano Mella
Annali dell'Università di Ferrara. Scienze Matematiche/Annali dell'Università di Ferrara. Sezione 7: Scienze matematiche, May 20, 2024
William Edge was born in Stockport on 8th November 1904. Both his parents were schoolteachers. Ed... more William Edge was born in Stockport on 8th November 1904. Both his parents were schoolteachers. Edge was educated at his local school, Stockport Grammar School. In 1923, he went to Cambridge where he studied mathematics at Trinity College. After graduating, Edge became a PhD student of Henry Baker. His dissertation generalized Luigi Cremona's results about ruled surfaces in the real projective space. In 1928, Edge became a research fellow in Trinity College. After 4 years he took a lectureship (assistant professorship) at the University of Edinburgh and remained there for the rest of his career. He was elected a Fellow of the Royal Society of Edinburgh two years later. Edge was promoted to reader (associate professor) in 1949. He became a full professor in 1969, six years prior his retirement in 1975. Edge never married, he had no children, he never drove a car, he was reluctant to travel, and he disdained radio and television. Apart from mathematics Edge loved hill walking, singing and playing the
arXiv (Cornell University), Dec 11, 2014
Let K be a number field and a i (t) ∈ K[t] polynomials of degree 2. We consider the family of ell... more Let K be a number field and a i (t) ∈ K[t] polynomials of degree 2. We consider the family of elliptic curves E t := y 2 = a 3 (t)x 3 + a 2 (t)x 2 + a 1 (t)x + a 0 (t) ⊂ A 2 xy (*) parametrized by t ∈ K. Our aim is to show that there are many values t ∈ K for which the corresponding elliptic curve E t has rank ≥ 1. Conjecturally this should hold for a positive proportion of them; see the survey [RS02]. We prove rank ≥ 1 for about the square root of all t ∈ K, listed by height. We are only interested in nontrivial families, when at least two of the curves E t are smooth, elliptic and not isomorphic to each other over K. Thus a 3 (t) is not identically 0 and not all the a i (t) are constant multiples of the same square (t − c) 2. We view the whole family as a single algebraic surface in A 3 xyt and look at the distribution of K-points. The resulting surface has degree 5 but its closure in P 3 is very singular. We prove the following. Theorem 1. Let k be any field of characteristic = 2 and a 0 (t),. .. , a 3 (t) ∈ k[t] polynomials of degree 2 giving a nontrivial family of elliptic curves. Then the surface S := y 2 = a 3 (t)x 3 + a 2 (t)x 2 + a 1 (t)x + a 0 (t) ⊂ A 3 xyt
Journal of the European Mathematical Society, Jan 4, 2022
A projective variety X ⊂ P N is h-identifiable if the generic element in its h-secant variety uni... more A projective variety X ⊂ P N is h-identifiable if the generic element in its h-secant variety uniquely determines h points on X. In this paper we propose an entirely new approach to study identifiability, connecting it to the notion of secant defect. In this way we are able to improve all known bounds on identifiability. In particular we give optimal bounds for some Segre and Segre-Veronese varieties and provide the first identifiability statements for Grassmann varieties.
Rendiconti Del Circolo Matematico Di Palermo, Jun 21, 2023
Two reduced projective schemes are said to be Cremona equivalent if there is a Cremona map that m... more Two reduced projective schemes are said to be Cremona equivalent if there is a Cremona map that maps one in the other. In this paper I revise some of the known results about Cremona equivalence and extend the main result of Mella and Polastri (Bull Lond Math Soc 41(1):89-93, 2009) [20] to reducible schemes. This allows to prove a very general contractibility result for union of rational subvarieties.
arXiv (Cornell University), Mar 8, 2017
Given a closed subvariety X in a projective space, the rank with respect to X of a point p in thi... more Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let W k be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci W k for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1, 3, 3), and curves. An intermediate result provides a lower bound on the dimension of any GL n orbit of a homogeneous form.
Rendiconti del Circolo Matematico di Palermo Series 2
Two reduced projective schemes are said to be Cremona equivalent if there is a Cremona map that m... more Two reduced projective schemes are said to be Cremona equivalent if there is a Cremona map that maps one in the other. In this paper I revise some of the known results about Cremona equivalence and extend the main result of Mella and Polastri (Bull Lond Math Soc 41(1):89–93, 2009) [20] to reducible schemes. This allows to prove a very general contractibility result for union of rational subvarieties.
arXiv (Cornell University), Oct 24, 2022
Let X ⊂ P hn+h−1 be an irreducible and non-degenerate variety of dimension n. The Bronowski's con... more Let X ⊂ P hn+h−1 be an irreducible and non-degenerate variety of dimension n. The Bronowski's conjecture predicts that X is h-identifiable if and only if the general (h − 1)tangential projection τ X h−1 : X P n is birational. In this paper we provide counterexamples to this conjecture. Building on the ideas that led to the counterexamples we manage to prove an amended version of the Bronowski's conjecture for a wide class of varieties and to reduce the identifiability problem for projective varieties to their secant defectiveness.
Crelle's Journal, Nov 12, 2017
A homogeneous polynomial of degree d in n + 1 variables is identifiable if it admits a unique add... more A homogeneous polynomial of degree d in n + 1 variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of d and n for which a general polynomial of degree d in n + 1 variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces.
arXiv (Cornell University), May 9, 1997
arXiv (Cornell University), May 27, 2013
In the last years the biregular automorphisms of the Deligne-Mumford's and Hassett's compactifica... more 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 M 0,n, and to the degenerations of Hassett's spaces obtained by allowing zero weights. Contents Introduction and Survey on the automorphisms of moduli spaces of curves 1 1. Kapranov's and Keel's spaces 10 2. Hassett's spaces with zero weights 13 References 15
arXiv (Cornell University), Jun 4, 2010
arXiv (Cornell University), Oct 27, 1994
Bollettino Della Unione Matematica Italiana, 1998
Sunto della tesi di dotorat
Trends in mathematics, 2023
Two birational projective varieties in P n are Cremona Equivalent if there is a birational modifi... more Two birational projective varieties in P n are Cremona Equivalent if there is a birational modification of P n mapping one onto the other. The minimal Cremona degree of X ⊂ P n is the minimal integer among all degrees of varieties that are Cremona Equivalent to X. The Cremona Equivalence and the minimal Cremona degree is well understood for subvarieties of codimension at least 2 while both are in general very subtle questions for divisors. In this note I compute the minimal Cremona degree of quartic surfaces in P 3. This allows me to show that any quartic surface of elliptic ruled type has non trivial stabilizers in the Cremona group.
Proceedings of the American Mathematical Society, 1998
Let X be an n-dimensional variety and E an ample vector bundle on X of rank e. We give a complete... more Let X be an n-dimensional variety and E an ample vector bundle on X of rank e. We give a complete classification of pairs (X, E), with X log terminal and e ≥ n such that K X + detE is not ample. The results we obtain were conjectured by Fujita, and recently by Zhang.
Trends in the history of science, 2016
In this paper we present an effective method for linearizing rational varieties of codimension at... more In this paper we present an effective method for linearizing rational varieties of codimension at least two under Cremona transformations, starting from a given parametrization. Using these linearizing Cremonas, we simplify the equations of secant and tangential varieties of some classical examples, including Veronese, Segre and Grassmann varieties. We end the paper by treating the special case of the Segre embedding of the n-fold product of projective spaces, where cumulant Cremonas, arising from algebraic statistics, appear as specific cases of our general construction.
arXiv (Cornell University), Mar 7, 2017
A tensor T , in a given tensor space, is said to be h-identifiable if it admits a unique decompos... more 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.
Bulletin of The London Mathematical Society, Jan 28, 2009
Let X be a projective variety of dimension r over an algebraically closed field. It is proven tha... more Let X be a projective variety of dimension r over an algebraically closed field. It is proven that two birational embeddings of X in P n , with n ≥ r + 2 are equivalent up to Cremona transformations of P n .
Sunto della tesi di dotorat
The aim of these talks is to give an overview to unirationality problems. I will discuss the beha... more The aim of these talks is to give an overview to unirationality problems. I will discuss the behaviour of unirationality in families and its relation with rational connectedness. Then I will concentrate on hypersurfaces and conic bundles. These special classes of varieties are a good place where to test different techniques and try to approach the unirationality problem via rational connectedness.
Uploads
Papers by Massimiliano Mella