arXiv subjects

Marco Andreatta

Publications and source records attributed to Marco Andreatta.

At least 19 recordsLinked to original sources

arXiv2025

Birational geometry of the twofold symmetric product of a Hirzebruch surface via secant maps

In this paper, extending some ideas of Fano, we study the birational geometry of the Hilbert scheme of 0-dimensional subschemes of length 2 of a rational normal scroll. This fourfold has three elementary contractions associated to the three faces of its nef cone. We study natural projective realizations of these contractions. In particular, given a smooth rational normal scroll $S_{a,b}$ of degree $r$ in ${\mathbb P}^{r+1}$ with $1 \leq a \leq b$ and a+b=r, i.e., $S_{a,b}$ is the relative Proj of the vector bundle $O_{{\mathbb P}^1}(a)\oplus O_{{\mathbb P}^1}(b)$ embedded in ${\mathbb P}^{r+1}$ with its O(1) line bundle (from an abstract viewpoint $S_{a,b}\cong {\mathbb F}_{b-a}$), we consider the subvariety $X_{a,b}$ of the Grassmannian $G(1,r+1)$ described by all lines that are secant or tangent to $S_{a,b}$. The variety $X_{a,b}$ is the image of some of the aforementioned contractions, it is smooth if a>1, and it is singular at a unique point if a=1. We compute the degree of $X_{a,b}$ and the local structure of the singularity of $X_{a,b}$ when a=1. Finally we discuss in some detail the case r=4, originally considered by Fano, because the smooth hyperplane sections of $X_{2,2}$ and $X_{1,3}$ are the Fano 3-folds that appear as number 16 in the Mori-Mukai list of Fano 3-folds with Picard number 2. We prove that any smooth hyperplane section of $X_{2,2}$ is also a hyperplane section of $X_{1,3}$, and we discuss the GIT-stability of the smooth hyperplane sections of $X_{1,3}$ where $G$ is the subgroup of the projective automorphisms of $X_{1,3}$ coming from the ones of $S_{1,3}.$

math.AG↗

arXiv2022

Fano's Last Fano

In 1949 Fano published his last paper on $3$-folds with canonical sectional curves. There he constructed and described a $3$-fold of the type $X^{22}_3$ in ${\mathbb P}^{13}$ with canonical curve section, which we like to call Fano's last Fano. We report on Fano's construction, providing various (in our opinion missing) proofs, in modern language and trying to use results and techniques available at that time. Then we construct Fano's with modern tools, in particular via the Hilbert scheme of zero cycles on a rational surface; as a consequence we easily point out the corresponding example in the Mori-Mukai classification.

math.AG↗

arXiv2018

Effective Adjunction Theory

Here we investigate the property of effectivity for adjoint divisors. Among others, we prove the following results: (i) A normal projective variety $X$ with at most canonical singularities is uniruled if and only if for each very ample Cartier divisor $H$ on $X$ we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$. (ii) Let $(X,L)$ be a polarized manifold of dimension $4$ and let $t$ be an integer with $t \ge 3$. If $K_X+tL$ is pseudo-effective, then $H^0(X, K_X+tL) \ne 0$.

math.AG↗

arXiv2017

Lifting Weighted Blow-ups

Let f: X -> Z be a local, projective, divisorial contraction between normal varieties of dimension n with Q-factorial singularities. Let $Y \subset X$ be a f-ample Cartier divisor and assume that f|Y: Y -> W has a structure of a weighted blow-up. We prove that f: X -> Z, as well, has a structure of weighted blow-up. As an application we consider a local projective contraction f: X -> Z from a variety X with terminal Q-factorial singularities, which contracts a prime divisor E to an isolated Q-factorial singularity $P\in Z$, such that $-(K_X + (n-3)L)$ is f-ample, for a f-ample Cartier divisor L on X. We prove that (Z,P) is a hyperquotient singularity and f is a weighted blow-up.

math.AG↗

arXiv2015

Local Fano-Mori contractions of high nef-value

Let $X$ be a variety with at most terminal $\mathbb Q$-factorial singularities of dimension $n$. We study local contractions $f:X\to Z$ supported by a $\mathbb Q$-Cartier divisor of the type $K_X+ τL$, where $L$ is an $f$-ample Cartier divisor and $τ\geq 0$ is a rational number. Equivalently, $f$ is a Fano-Mori contraction associated to an extremal face in $\overline {NE(X)}_{K_X+τL = 0}$; these maps naturally arise in the context of the minimal model program. We prove that, if $τ> (n-3) >0$, the general element $X' \in |L|$ is a variety with at most terminal singularities. Then we apply this to characterize, via an inductive argument, some birational contractions as above with $τ> (n-3)\geq 0$.

math.AG↗

arXiv2013

Fano-Mori contractions of high length on projective varieties with terminal singularities

Let X be a projective variety with terminal singularities and let L be an ample Cartier divisor on X. We prove that if f is a birational contraction associated to an extremal ray $ R \subset \bar {NE(X)}$ such that R.(K_X+(n-2)L)<0, then f is a weighted blow-up of a smooth point. We then classify divisorial contractions associated to extremal rays R such that R.(K_X+rL)<0, where r is a non-negative rational number, and the fibres of f have dimension less or equal to r+1.

math.AG↗

arXiv2013

4-dimensional symplectic contractions

Local symplectic contractions are resolutions of singularities which admit symplectic forms. Four dimensional symplectic contractions are (relative) Mori Dream Spaces. In particular, any two such resolutions of a given singularity are connected by a sequence of Mukai flops. We discuss the cone of movable divisors on such a resolution; its faces are determined by curves whose loci are divisors, we call them essential curves. The movable cone is divided into nef chambers which are related to different resolutions; this subdivision is determined by classes of 1-cycles. We also study schemes parametrizing minimal essential curves and show that they are resolutions, possibly non-minimal, of surface Du Val singularities. Some examples, with an exhaustive description, are provided.

math.AG↗

arXiv2012

Minimal Model Program with scaling and adjunction theory

Let (X,L) be a quasi polarized pairs, i.e. X is a normal complex projective variety and L is a nef and big line bundle on it. We study, up to birational equivalence, the positivity (nefness) of the adjoint bundles K_X + rL for high rational number r. For this we run a Minimal Model Program with scaling relative to the divisor K_X +rL. We give some applications, namely the classification up to birational equivalence of quasi polarized pairs with sectional genus 0,1 and of embedded projective varieties X < P^N with degree smaller than 2codim(X) +2.

math.AG↗

arXiv2009

On the Kummer construction

We discuss a generalization of Kummer construction which, on the base of an integral representation of a finite group and local resolution of its quotient, produces a higher dimensional variety with trivial canonical class. As an application we compute cohomology of some generalized Kummer varieties.

math.AG↗

arXiv2005

Fano manifolds with long extremal rays

Let X be a Fano manifold of pseudoindex i_X whose Picard number is at least two and let R be an extremal ray of X with exceptional locus Exc(R). We prove an inequality which bounds the length of R in terms of i_X and of the dimension of Exc(R) and we investigate the border cases. In particular we classify Fano manifolds X of pseudoindex i_X obtained blowing up a smooth variety Y along a smooth subvariety T such that dim T < i_X.

math.AG↗

arXiv2003

Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex i_X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i_X-1)\le n. We prove that the conjecture holds if: a) X has pseudoindex i_X \ge \frac{n+3}{3} and either has a fiber type extremal contraction or does not have small extremal contractions b) X has dimension five.

math.AG↗

arXiv2003

Characterization theorems for the projective space and vector bundle adjunction

We consider some conditions under which a smooth projective variety X is actually the projective space. We also extend to the case of positive characteristic some results in the theory of vector bundle adjunction. We use methods and techniques of the so called Mori theory, in particular the study of rational curves on projective manifolds.

math.AG↗

arXiv2000

Special rays in the Mori cone of a projective variety

Let $X$ be a smooth $n$-dimensional projective variety over an algebraically closed field $k$ such that $K_X$ is not nef. We give a characterization of non nef extremal rays of $X$ of maximal length (i.e of length $n-1$); in the case of $\ch(k) = 0$ we also characterize non nef rays of length $n-2$.

math.AG↗

arXiv1998

Actions of Linear Algebraic Groups on Projective Manifolds and Minimal Model Program

Let X be a smooth projective variety of dimension n on which a simple Lie group G acts regularly and non trivially. Then X is not minimal in the sense of the Minimal Model Program. In the paper we work out a classification of X via the Minimal Model Program under the assumption that the dimension of X is small with the respect to the dimension of G. More precisely we classify all such X with n smaller or equal to (r_G +1), where r_G is the minimum codimension of the maximal parabolic subgroup of G (for instance r_{SL(m)}= m-1). We consider also the case when G = SL(3) and X is a smooth 4-fold on which G acts with an open orbit.

math.AG↗

arXiv1998

Ample vector bundles with sections vanishing on special varieties

Let E be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section $s \in Γ(\cal E)$ whose zero locus Z = (s = 0) is a smooth submanifold of the expected dimension dim X - r: = n -r. Assume that Z is not minimal; we investigate the hypothesis under which the extremal contractions of Z can be lifted to X. Finally we study in detail the cases in which Z is a scroll, a quadric bundle or a del Pezzo fibration.

math.AG↗

arXiv1997

Moishezon Manifolds

Let X be a compact Moishezon manifold which becomes projective after blowing up a smooth subvariety $Y \subset X$. We assume also that there exists a proper map $ρ:X \to X'$ onto a projective variety X' with $ρ(Y)$ a point, such that $Pic(X/X') = \Z$ and $K_X$ is $ρ$-big. We prove some inequalities between the dimensions of Y and X and we construct examples which shows the optimality of the inequalities. Then we discuss some differential geometry properties of these examples which lead to a conjecture.

alg-geom↗

arXiv1996

A view on contractions of higher dimensional varieties

In this paper we discuss some recent results about extremal contractions of complex algebraic varieties. These are proper surjective maps, $ϕ: X\longrightarrow Z$, of normal varieties with connected fibers such that $X$ has mild singularities (in particular $X$ is $Q$-Gorenstein) and the anticanonical divisor of $X$ is $ϕ$ ample. It will appear on the Proceedings of AMS-SRI on Algebraic Geometry, Santa Cruz 1995.

alg-geom↗

arXiv1996

On Contractions of smooth varieties

Let $\f: X \ra Z$ be a proper surjective map from a smooth complex manifold $X$ onto a normal variety $Z$. If $\f$ has connected fibers and $-K_X$ is $\f$-ample then $\f$ is called a good contraction. In the present paper we study good contractions, fibers of which have dimension less or equal than two: after describing possible two dimensional isolated fibers we discuss their scheme theoretic structure and the geometry of $\f:X\ra Z$ nearby such a fiber. If $dimX=4$ and $\f$ is birational with an isolated 2 dimensional fiber then we obtain a complete description of $\f$. We provide also a description of a 4 dimensional conic fibration with an isolated fiber which is either a plane or a quadric. We construct pertinent examples.

alg-geom↗