arXiv subjects

Andrea Luigi Tironi

Publications and source records attributed to Andrea Luigi Tironi.

18 recordsLinked to original sources

arXiv2022

On skew partial derivatives and a Hermite-type interpolation problem

Let $\mathcal{R}:=\mathbb{F}[{\bf x};σ,δ]$ be a multivariate skew polynomial ring over a division ring $\mathbb{F}$. In this paper, we introduce the notion of right and left $(σ,δ)$-partial derivatives of polynomials in $\mathcal{R}$ and we prove some of their main properties. As an application of these results, we solve in $\mathcal{R}$ a Hermite-type multivariate skew polynomial interpolation problem. The main technical tools and results used here are of constructive type, showing methods and algorithms to construct a polynomial in $\mathcal{R}$ which satisfies the above Hermite-type interpolation problem and its relative Lagrange-type version.

math.RA↗

arXiv2022

Resultants of skew polynomials over division rings

Let $\mathbb{F}$ be a division ring. In this paper, we extent some of the main well-known results about the resultant of two univariate polynomials to the more general context of an Ore extension $\mathbb{F}[x;σ,δ]$. Finally, some algorithms and Magma programs are given as a numerical application of the main theoretical results of this paper.

math.RA↗

arXiv2022

Some Fano manifolds whose Hilbert polynomial is totally reducible over $\mathbb Q$

Let $(X,L)$ be any Fano manifold polarized by a positive multiple of its fundamental divisor $H$. The polynomial defining the Hilbert curve of $(X,L)$ boils down to being the Hilbert polynomial of $(X,H)$, hence it is totally reducible over $\mathbb C$; moreover, some of the linear factors appearing in the factorization have rational coefficients, e.g. if $X$ has index $\geq 2$. It is natural to ask when the same happens for all linear factors. Here the total reducibility over $\mathbb Q$ of the Hilbert polynomial is investigated for three special kinds of Fano manifolds: Fano manifolds of large index, toric Fano manifolds of low dimension, and Fano bundles of low coindex.

math.AG↗

arXiv2020

On a Homma-Kim conjecture for nonsingular hypersurfaces

Let $X^n$ be a nonsingular hypersurface of degree $d\geq 2$ in the projective space $\mathbb{P}^{n+1}$ defined over a finite field $\mathbb{F}_q$ of $q$ elements. We prove a Homma-Kim conjecture on a upper bound about the number of $\mathbb{F}_q$-points of $X^n$ for $n=3$, and for any odd integer $n\geq 5$ and $d\leq q$.

math.AG↗

arXiv2019

Lower and upper bounds for some generalized arcs

Let $\mathbb{F}_q$ be a field with $q$ elements. In this note, we study some generalized arcs, that is, sets of $\mathbb{F}_q$-points in the projective plane $\mathbb{P}^2(\mathbb{F}_q)$ such that no six of them are on a conic. First, we consider the geometric configurations of such generalized arcs for small values of $q$ and then we give some upper and lower bounds for the cardinality of complete generalized arcs, i.e. generalized arcs which are not contained in a bigger one.

math.AG↗

arXiv2019

The Hilbert curve of a 4-dimensional scroll with a divisorial fiber

In dimension $n = 2m-2 \geq 4$ adjunction theoretic scrolls over a smooth $m$-fold may not be classical scrolls, due to the existence of divisorial fibers. A $4$-dimensional scroll $(X,L)$ over $\mathbb P^3$ of this type is considered, and the equation of its Hilbert curve $Γ$ is determined in two ways, one of which relies on the fact that $(X,L)$ is at the same time a classical scroll over a threefold $Y \not=\mathbb P^3$. It turns out that $Γ$ does not perceive divisorial fibers. The equation we obtain also shows that a question raised in a previous article by Beltrametti, Lanteri and Sommese, has negative answer in general for non-classical scrolls over a $3$-fold. More precisely, the answer for $(X,L)$ is negative or positive according to whether $(X,L)$ is regarded as an adjunction theoretic scroll or as a classical scroll; in other words, it is the answer to this question to distinguish between the existence of jumping fibers or not.

math.AG↗

arXiv2018

Generalized polarized manifolds with low second class

On a smooth complex projective variety $X$ of dimension $n$, consider an ample vector bundle $\mathcal{E}$ of rank $r \leq n-2$ and an ample line bundle $H$. A numerical character $m_2=m_2(X,\mathcal{E},H)$ of the triplet $(X,\mathcal{E},H)$ is defined, extending the well-known second class of a polarized manifold $(X,H)$, when either $n=2$ or $H$ is very ample. Under some additional assumptions on $\mathcal{F}: = \mathcal{E} \oplus H^{\oplus (n-r-2)}$, triplets $(X,\mathcal{E},H)$ as above whose $m_2$ is small with respect to the invariants $d:=c_{n-2}(\mathcal{F})H^2$ and $g:=1+\frac{1}{2}\big(K_X + c_1(\mathcal{F})+H\big) \cdot c_{n-2}(\mathcal{F}) \cdot H$ are studied and classified.

math.AG↗

arXiv2018

A note on the duals of skew constacyclic codes

Let $\mathbb{F}_q$ be a finite field with $q$ elements and denote by $θ: \mathbb{F}_q\to\mathbb{F}_q$ an automorphism of $\mathbb{F}_q$. In this paper, we deal with skew constacyclic codes, that is, linear codes of $\mathbb{F}_q^n$ which are invariant under the action of a semi-linear map $Φ_{α,θ}:\mathbb{F}_q^n\to\mathbb{F}_q^n$, defined by $Φ_{α,θ}(a_0,...,a_{n-2}, a_{n-1}):=(αθ(a_{n-1}),θ(a_0),...,θ(a_{n-2}))$ for some $α\in\mathbb{F}_q\setminus\{0\}$ and $n\geq 2$. In particular, we study some algebraic and geometric properties of their dual codes and we give some consequences and research results on $1$-generator skew quasi-twisted codes and on MDS skew constacyclic codes.

cs.IT↗

arXiv2018

Hilbert curve characterizations of some relevant polarized manifolds

Hilbert curves of special varieties like Fano manifolds of low coindex as well as fibrations having such a manifold as general fiber, endowed with appropriate polarizations, are investigated. In particular, all most relevant varieties arising in adjunction theory are characterized in terms of their Hilbert curves.

math.AG↗

arXiv2018

On two upper bounds for hypersurfaces involving a Thas' invariant

Let $X^n$ be a hypersurface in $\mathbb{P}^{n+1}$ with $n\geq 1$ defined over a finite field $\mathbb{F}_q$ of $q$ elements. In this note, we classify, up to projective equivalence, hypersurfaces $X^n$ as above which reach two elementary upper bounds for the number of $\mathbb{F}_q$-points on $X^n$ which involve a Thas' invariant.

math.AG↗

arXiv2016

Hypersurfaces achieving the Homma-Kim bound

Let $X$ be a hypersurface in $\mathbb{P}^N$ with $N\geq 3$ defined over a finite field. The main result of this note is the classification, up to projective equivalence, of hypersurfaces $X$ as above without a linear component when the number of their rational points achieves the Homma-Kim bound.

math.AG↗

arXiv2015

On del Pezzo elliptic varieties of degree $\leq 4$

\special{html: } Let $Y$ be a del Pezzo variety of degree $d\leq 4$ and dimension $n\geq 3$, let $H$ be an ample class such that $-K_Y=(n-1)H$ and let $Z\subset Y$ be a $0$-dimensional subscheme of length $d$ such that the subsystem of elements of $|H|$ with base locus $Z$ gives a rational morphism $π_Z\colon Y\dashrightarrow{\mathbb P}^{n-1}$. Denote by $π\colon X\to {\mathbb P}^{n-1}$ the elliptic fibration obtained by resolving the indeterminacy locus of $π_Z$. Extending the results of [arXiv:1305.3340] we study the geometry of the variety $X$ and we prove that the Mordell-Weil group of $π$ is finite if and only if the Cox ring of $X$ is finitely generated.

math.AG↗

arXiv2015

Some properties of skew codes over finite fields

After recalling the definition of codes as modules over skew polynomial rings, whose multiplication is defined by using an automorphism and a derivation, and some basic facts about them, in the first part of this paper we study some of their main algebraic and geometric properties. Finally, for module skew codes constructed only with an automorphism, we give some BCH type lower bounds for their minimum distance.

cs.IT↗

arXiv2014

Varieties of Picard rank one as components of ample divisors

Let $\mathcal{V}$ be an integral normal complex projective variety of dimension $n\geq 3$ and denote by $\mathcal{L}$ an ample line bundle on $\mathcal{V}$. By imposing that the linear system $|\mathcal{L}|$ contains an element $A=A_{1}+...+A_{r}, r\geq 1$, where all the $A_{i}$'s are distinct effective Cartier divisors with Pic$(A_i)=\mathbb{Z}$, we show that such a $\mathcal{V}$ is as special as the components $A_i$ of $A\in |\mathcal{L}|$. After making a list of some consequences about the positivity of the components $A_i$, we characterize pairs $(\mathcal{V}, \mathcal{L})$ as above when either $A_1\cong\mathbb{P}^{n-1}$ and Pic$(A_j)=\mathbb{Z}$ for $j=2,...,r,$ or $\mathcal{V}$ is smooth and each $A_i$ is a variety of small degree with respect to $[H_i]_{A_i}$, where $[H_i]_{A_i}$ is the restriction to $A_i$ of a suitable line bundle $H_i$ on $\mathcal{V}$.

math.AG↗

arXiv2013

Dual codes of product semi-linear codes

Let $\mathbb{F}_q$ be a finite field with $q$ elements and denote by $θ: \mathbb{F}_q\to\mathbb{F}_q$ an automorphism of $\mathbb{F}_q$. In this paper, we deal with linear codes of $\mathbb{F}_q^n$ invariant under a semi-linear map $T:\mathbb{F}_q^n\to\mathbb{F}_q^n$ for some $n\geq 2$. In particular, we study three kind of their dual codes, some relations between them and we focus on codes which are products of module skew codes in the non-commutative polynomial ring $\mathbb{F}_q[X,θ]$ as a subcase of linear codes invariant by a semi-linear map $T$. In this setting we give also an algorithm for encoding, decoding and detecting errors and we show a method to construct codes invariant under a fixed $T$.

cs.IT↗

arXiv2013

On cubic elliptic varieties

Let X->P^(n-1) be an elliptic fibration obtained by resolving the indeterminacy of the projection of a cubic hypersurface Y of P^(n+1) from a line L not contained in Y. We prove that the Mordell-Weil group of the elliptic fibration is finite if and only if the Cox ring of X is finitely generated. We also provide a presentation of the Cox ring of X when it is finitely generated.

math.AG↗

arXiv2012

Normal projective varieties of degree 5

We list the irreducible reduced and not degenerate normal projective varieties $X\subset\mathbb{P}^N$ of dimension $n$ and degree five defined over an algebraically closed field $k$ of char$(k) = 0$. In the smooth case, or when $n = 2$, we give also classification results for any algebraically closed field $k$ of char$(k) \geq 0$.

math.AG↗

arXiv2011

Nefness of adjoint bundles for ample vector bundles of corank 3

Let $\mathcal{E}$ be an ample vector bundle of rank $r\geq 2$ on a smooth complex projective variety $X$ of dimension $n$. The aim of this paper is to describe the structure of pairs $(X,\mathcal{E})$ as above whose adjoint bundles $K_X+\det\mathcal{E}$ are not nef for $r=n-3$. Furthermore, we give some immediate consequences of this result in adjunction theory.

math.AG↗