arXiv subjects

Antonio Lanteri

Publications and source records attributed to Antonio Lanteri.

16 recordsLinked to original sources

arXiv2023

Triple solids and scrolls

Let $Y$ be a smooth projective variety of dimension $n \geq 2$ endowed with a finite morphism $ϕ:Y \to \mathbb P^n$ of degree $3$, and suppose that $Y$, polarized by some ample line bundle, is a scroll over a smooth variety $X$ of dimension $m$. Then $n \leq 3$ and either $m=1$ or $2$. When $m=1$, a complete description of the few varieties $Y$ satisfying these conditions is provided. When $m=2$, various restrictions are discussed showing that in several instances the possibilities for such a $Y$ reduce to the single case of the Segre product $\mathbb P^2 \times \mathbb P^1$. This happens, in particular, if $Y$ is a Fano threefold as well as if the base surface $X$ is $\mathbb P^2$.

math.AG↗

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

Rational conic fibrations of sectional genus two

Polarized rational surfaces $(X, \mathcal L)$ of sectional genus two ruled in conics are studied. When they are not minimal, they are described as the blow-up of $\mathbb F_1$ at some points lying on distinct fibers. Ampleness and very ampleness of $\mathcal L$ are studied in terms of their location. When $\mathcal L$ is very ample and there is a line contained in $X$ and transverse to the fibers, the conic fibrations $(X, \mathcal L)$ are classified and a related property concerned with the inflectional locus is discussed.

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

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↗

arXiv2011

Geometry of rays-positive manifolds

Let M be a smooth complex projective variety and let L be a line bundle on it. Rays-positive manifolds, namely pairs (M,L) such that L is numerically effective and L\cdotR > 0 for all extremal rays R on M, are studied. Several illustrative examples and some applications are provided. In particular, projective varieties with crepant singularities and of small degree with respect to the codimension are classified, and the non-negativity of the sectional genus g(M,L) is proven, describing as well the pairs with g(M,L) = 0,1.

math.AG↗

arXiv2010

Inflectional loci of scrolls II

Let $X\subset \mathbb P^N$ be a scroll over a $m$-dimensional variety $Y$. We find the locally free sheaves on $X$ governing the osculating behavior of $X$, and, under certain dimension assumptions, we compute the cohomology class and the degree of the inflectional locus of $X$. The case $m=1$ was treated in \cite{LMP}. Here we treat the case $m\ge 2$, which is more complicated for at least two reasons: the expression for the osculating sheaves and the computations of the class of the inflectional locus become more complex, and the dimension requirements needed to ensure validity of the formulas are more severe.

math.AG↗

arXiv2008

Low dimensional discriminant loci and scrolls

Smooth complex polarized varieties $(X,L)$ with a vector subspace $V \subseteq H^0(X,L)$ spanning $L$ are classified under the assumption that the locus ${\Cal D}(X,V)$ of singular elements of $|V|$ has codimension equal to $\dim(X)-i$, $i=3,4,5$, the last case under the additional assumption that $X$ has Picard number one. In fact it is proven that this codimension cannot be $\dim(X)-4$ while it is $\dim(X)-3$ if and only if $(X,L)$ is a scroll over a smooth curve. When the codimension is $\dim(X)-5$ and the Picard number is one only the Plücker embedding of the Grassmannian of lines in $\Bbb P^4$ or one of its hyperplane sections appear. One of the main ingredients is the computation of the top Chern class of the first jet bundle of scrolls and hyperquadric fibrations. Further consequences of these computations are also provided.

math.AG↗

arXiv2008

Ample subvarieties and rationally connected fibrations

Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibration.

math.AG↗

arXiv2007

Osculating properties of decomposable scrolls

Osculating spaces of decomposable scrolls (of any genus and not necessarily normal)are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra components of the second discriminant locus - deriving from flexes - are investigated and a new class of uninflected surface scrolls is presented and characterized. Further properties related to osculation are discussed for (not necessarily decomposable) scrolls.

math.AG↗

arXiv2007

Inflectional loci of scrolls

Let $X\subset \mathbb P^N$ be a scroll over a smooth curve $C$ and let $Ł=\mathcal O_{\mathbb P^N}(1)|_X$ denote the hyperplane bundle. The special geometry of $X$ implies that some sheaves related to the principal part bundles of $Ł$ are locally free. The inflectional loci of $X$ can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls.

math.AG↗

arXiv2007

Higher Order Bad Loci

Zero-schemes on smooth complex projective varieties, forcing all elements of ample and free linear systems to be reducible are studied. Relationships among the minimal length of such zero-schemes, the positivity of the line bundle associated with the linear system, and the dimension of the variety are established. A generalization to higher dimension subschemes is studied in the last section.

math.AG↗

arXiv2007

Discriminant loci of ample and spanned line bundles

Let $(X,L,V)$ be a triplet where $X$ is an irreducible smooth complex projective variety, $L$ is an ample and spanned line bundle on $X$ and $V\subseteq H^0(X,L)$ spans $L$. The discriminant locus $\Cal D(X,V) \subset |V|$ is the algebraic subset of singular elements of $|V|$. We study the components of $\Cal D(X,V)$ in connection with the jumping sets of $(X,V)$, generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of $(X,L,V)$) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.

math.AP↗

arXiv2004

Bad loci of free linear systems

The bad locus of a base-point free linear system L on a normal complex projective variety X is defined as the subset B(L) of X of points that are not contained in any irreducible and reduced member of L. In this paper we provide a geometric description of such locus in terms of the morphism defined by L. In particular, assume that the dimension of X is at least 2, and that L is the complete linear system associated to an ample and spanned line bundle. It is known that in this case B(L) is empty unless X is a surface. Then we prove that, when the latter occurs, B(L) is not empty if and only if L defines a morphism onto a two dimensional cone, in which case B(L) is the inverse image of the vertex of the cone.

math.AG↗

arXiv2003

Peculiar Loci of Ample and Spanned Line Bundles

The bad locus and the rude locus of an ample and base point free linear system on a smooth complex projective variety are introduced and studied. The bad locus is defined as the set of points that force divisors through them to be reducible. The rude locus is defined as the set of points such that divisors that are singular at them are forced to be reducible. The existence of a nonmempty bad locus is shown to be exclusively a two dimensional phenomenon. Polarized surfaces of small degree, or whose degree is the square of a prime, with nonempty bad loci are completely classified. Several explicit examples are offered to describe the variety of behaviors of the two loci.

math.AG↗