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Gianluca Occhetta

Publications and source records attributed to Gianluca Occhetta.

At least 19 recordsLinked to original sources
arXiv2026

A two-step approach to Chow quotients

The Chow quotient of a projective variety by the action of a complex torus is known to have a very complicated geometry, even in the case of simple varieties, such as rational homogeneous varieties. In this paper we propose an approach in which the geometry of the Chow quotient is encoded in a projective toric variety and a finite subgroup of its birational automorphisms. We then illustrate how to apply our strategy in the case of some particular rational homogeneous varieties.

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arXiv2025

Constructing geometric realizations of birational maps between Mori Dream Spaces

We construct geometric realizations -- projective algebraic versions of cobordisms -- for birational maps between Mori Dream Spaces. We show that these geometric realizations are Mori Dream Spaces, as well, and that they can be constructed so that they induce factorizations of the original birational maps as compositions of wall-crossings. In the case of toric birational maps between normal $\mathbb{Q}$-factorial, projective toric varieties, we provide several SageMath functions to work with $\mathbb{C}^*$-actions and birational geometry; in particular we show how to explicitly construct a moment polytope of a toric geometric realization. Moreover, by embedding Mori Dream Spaces in toric varieties, we obtain geometric realizations of birational maps of Mori Dream Spaces as restrictions of toric geometric realizations. We also provide examples and discuss when a geometric realization is Fano.

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arXiv2025

Chow quotients of $\mathbb{C}^*$-actions

Given an action of the one-dimensional torus on a projective variety, the associated Chow quotient arises as a natural parameter space of invariant $1$-cycles, which dominates the GIT quotients of the variety. In this paper we explore the relation between the Chow and the GIT quotients of a variety, showing how to construct explicitly the former upon the latter via successive blowups under suitable assumptions. We also discuss conditions for the smoothness of the Chow quotient, and present some examples in which it is singular.

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arXiv2025

Chow quotients of ${\mathbb C}^*$-actions on convex varieties

In this paper we study the Chow quotient ${\mathcal C}X$ of a convex variety $X$ of Picard number one by the action of a one dimensional torus having no non-trivial finite isotropy. Examples of these actions can be found in the rational homogeneous framework. We prove that the subvariety of ${\mathcal C}X$ parametrizing reducible torus-invariant cycles is a simple normal crossing divisor, we compute the Nef and Mori cones of ${\mathcal C}X$, and its anticanonical divisor.

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arXiv2024

Characterizing rational homogeneous spaces via $\mathbb{C}^*$-actions

We study smooth varieties of Picard number one admitting a special dominating family of rational curves and an equalized $\mathbb{C}^*$-action. In particular we show that $X$ is a smooth variety of Picard number one with nef tangent bundle admitting an equalized $\mathbb{C}^*$-action with an isolated extremal fixed point if and only if $X$ is an irreducible Hermitian symmetric space.

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arXiv2024

Morphisms between Grassmannians, II

Denote by $\mathbb G(k,n)$ the Grassmannian of linear subspaces of dimension $k$ in $\mathbb P^n$. We show that, if $φ:\mathbb G(l,n) \to \mathbb G(k,n)$ is a non constant morphism and $l \not=0,n-1$ then $l=k$ or $l=n-k-1$ and $φ$ is an isomorphism.

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arXiv2023

Geometric realizations of birational transformations via $\mathbb{C}^*$-actions

In this paper we study varieties admitting torus actions as geometric realizations of birational transformations. We present an explicit construction of these geometric realizations for a particular class of birational transformations, and study some of their geometric properties, such as their Mori, Nef and Movable cones.

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arXiv2023

Maximal disjoint Schubert cycles in rational homogeneous varieties

In this paper we study properties of the Chow ring of rational homogeneous varieties of classical type, more concretely, effective zero divisors of low codimension, and a related invariant called effective good divisibility. This information is then used to study the question of (non)existence of nonconstant maps among these varieties, generalizing previous results for projective spaces and Grassmannians.

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arXiv2022

Morphisms between Grassmannians

Denote by $\mathbb G(k,n)$ the Grassmannian of linear subspaces of dimension $k$ in $\mathbb P^n$. We show that if $n>m$ then every morphism $φ: \mathbb G(k,n) \to \mathbb G(l,m)$ is constant.

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arXiv2021

Rational homogeneous spaces as geometric realizations of birational transformations

A geometric realization of a birational map $ψ$ among two complex projective varieties is a variety $X$ endowed with a $\mathbb{C}^*$-action inducing $ψ$ as the natural birational map among two extremal geometric quotients. In this paper we study geometric realizations of some classic birational maps --inversion maps, special Cremona transformations, special birational transformations of type $(2,1)$--, by considering $\mathbb{C}^*$-actions on certain rational homogeneous spaces and their subvarieties.

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arXiv2021

Small bandwidth ${\mathbb C}^*$-actions and birational geometry

In this paper we study smooth projective varieties and polarized pairs with an action of a one dimensional complex torus. As a main tool, we define birational geometric counterparts of these actions, that, under certain assumptions, encode the information necessary to reconstruct them. In particular, we consider some cases of actions of low complexity -- measured in terms of two invariants of the action, called bandwidth and bordism rank -- and discuss how they are determined by well known birational transformations, namely Atiyah flips and Cremona transformations.

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arXiv2021

High rank torus actions on contact manifolds

We prove LeBrun--Salamon conjecture in the following situation: if $X$ is a contact Fano manifold of dimension $2n+1$ whose group of automorphisms is reductive of rank $\geq \max(2,(n-3)/2)$ then $X$ is the adjoint variety of a simple group. The rank assumption is fulfilled not only by the three series of classical linear groups but also by almost all the exceptional ones.

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arXiv2021

Small modifications of Mori dream spaces arising from ${\mathbb C}^*$-actions

We link small modifications of projective varieties with a ${\mathbb C}^*$-action to their GIT quotients. Namely, using flips with centers in closures of Białynicki-Birula cells, we produce a system of birational equivariant modifications of the original variety, which includes those on which a quotient map extends from a set of semistable points to a regular morphism. The structure of the modifications is completely described for the blowup along the sink and the source of smooth varieties with Picard number one with a ${\mathbb C}^*$-action which has no finite isotropy for any point. Examples can be constructed upon homogeneous varieties with a ${\mathbb C}^*$-action associated to short grading of their Lie algebras.

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arXiv2020

Manifolds with two projective bundle structures

In this paper we classify varieties of Picard number two having two projective bundle structures of any relative dimension, under the assumption that these structures are mutually uniform. As an application we prove the Campana--Peternell conjecture for varieties of Picard number one admitting $\mathbb C^*$-actions of a certain kind.

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arXiv2019

Nestings of rational homogeneous varieties

In this paper we study the existence of sections of universal bundles on rational homogeneous varieties -- called nestings -- classifying them completely in the case in which the Lie algebra of the automorphism group of the variety is simple of classical type. In particular we show that, under this hypothesis, nestings do not exist unless there exists a proper algebraic subgroup of the automorphism group acting transitively on the base variety.

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arXiv2019

Deformation of Bott-Samelson varieties and variations of isotropy structures

In the framework of the problem of characterizing complete flag manifolds by their contractions, the complete flags of type $F_4$ and $G_2$ satisfy the property that any possible tower of Bott-Samelson varieties dominating them birationally deforms in a nontrivial moduli. In this paper we illustrate the fact that, at least in some cases, these deformations can be explained in terms of automorphisms of Schubert varieties, providing variations of certain isotropic structures on them. As a corollary, we provide a unified and completely algebraic proof of the characterization of complete flag manifolds in terms of their contractions.

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