arXiv subjects

Eleonora A. Romano

Publications and source records attributed to Eleonora A. Romano.

12 recordsLinked to original sources

arXiv2025

K-polystability of Fano 4-folds with large Lefschetz defect

In this paper, we investigate K-polystability on smooth complex Fano 4-folds with Lefschetz defect at least 2, focusing on the case of Lefschetz defect 3 and on Casagrande-Druel Fano 4-folds with Lefschetz defect 2. We show that exactly 5 of the 19 families of Fano 4-folds with Lefschetz defect 3 are K-polystable. Moreover, among 175 families of Casagrande-Druel Fano 4-folds with Lefschetz defect 2, we prove that 5 are K-polystable, while 132 are K-unstable.

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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

Mori Dream Bonds and $\mathbb{C}^*$-actions

We construct a correspondence between Mori dream regions arising from small modifications of normal projective varieties and $\mathbb{C}^*$-actions on polarized pairs which are bordisms. Moreover, we show that the Mori dream regions constructed in this way admit a chamber decomposition on which the models are the geometric quotients of the $\mathbb{C}^*$-action. In addition we construct, from a given $\mathbb{C}^*$-action on a polarized pair for which there exist at least two admissible geometric quotients, a $\mathbb{C}^*$-equivariantly birational $\mathbb{C}^*$-variety, whose induced action is a bordism, called the pruning of the variety.

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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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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

Toric non-equalized flips associated to $\mathbb{C}^*$-actions

Starting from $\mathbb{C}^*$-actions on complex projective varieties, we construct and investigate birational maps among the corresponding extremal fixed point components. We study the case in which such birational maps are locally described by toric flips, either of Atiyah type or so called non-equalized. We relate this notion of toric flip with the property of the action being non-equalized. Moreover, we find explicit examples of rational homogeneous varieties admitting a $\mathbb{C}^*$-action whose weighted blow-up at the extremal fixed point components gives a birational map among two projective varieties that is locally a toric non-equalized flip.

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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

Classification of Fano 4-folds with Lefschetz defect 3 and Picard number 5

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. If X contains a prime divisor D with rho(X)-rho(D)>2, then either X is a product of del Pezzo surfaces, or rho(X)=5 or 6. In this setting, we completely classify the case where rho(X)=5; there are 6 families, among which one is new. We also deduce the classification of Fano 4-folds with rho(X)>4 with an elementary divisorial contraction sending a divisor to a curve.

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arXiv2020

Adjunction for varieties with a $\mathbb{C}^*$ action

Let $X$ be a complex projective manifold, $L$ an ample line bundle on $X$, and assume that we have a $\mathbb{C}^*$ action on $(X,L)$. We classify such triples $(X,L,\mathbb{C}^*)$ for which the closure of a general orbit of the $\mathbb{C}^*$ action is of degree $\leq 3$ with respect to $L$ and, in addition, the source and the sink of the action are isolated fixed points, and the $\mathbb{C}^*$ action on the normal bundle of every fixed point component has weights $\pm 1$. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension $11$ and $13$ are homogeneous if their group of automorphisms is reductive of rank $\geq 2$.

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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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