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Antonio Laface

Publications and source records attributed to Antonio Laface.

At least 19 recordsLinked to original sources
arXiv2026

Fujita freeness for projectivized toric vector bundles

Let $X$ be a smooth projective toric variety of dimension $n\geq1$ over an algebraically closed field of characteristic zero, let $\mathcal E$ be a toric vector bundle of rank $r\geq2$, and let $\pi\colon Y=\mathbb P_X(\mathcal E)\to X$ be the projective bundle of one-dimensional quotients. Write an ample line bundle on $Y$ as $A=\mathcal O_Y(a)\otimes\pi^*L$, with $a\geq1$. We record a blow-up argument proving that $K_Y+mA$ is globally generated whenever an integer $m$ satisfies $ma\geq r$ and $m\delta(A)>n$, where $\delta(A)$ is a positive integer obtained from the degrees of $A$ on the invariant quotient sections over the torus-invariant curves of $X$. In particular, $K_Y+mA$ is globally generated for $m\geq n+1$ and $ma\geq r$. Consequently every projectivized toric vector bundle satisfies Fujita's freeness conjecture. The uniform bound is sharp. We also formulate the result as a global-generation theorem for adjoint symmetric powers of $\mathcal E$ and explain its relation with the Seshadri-constant results of Hering--Musta\c{t}\u{a}--Payne and Fulger--Murayama. ChatGPT (OpenAI) was used to assist with mathematical discussion, language, and bibliographic searches.

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arXiv2026

Multiplicity-One Cox Rings of Toric Point Blow-Ups

Let $X$ be a complete toric variety and let $e$ be the identity of its open torus. The Cox ring of ${\rm Bl}_eX$ is described by saturated powers of the toric-point lattice ideal $I_X$. We give a finite criterion for generation in Rees multiplicity one in terms of analytic spreads of projected lattice ideals and zero divisors in the associated graded ring. We apply this criterion first to fake weighted projective spaces, proving that multiplicity-one generation is equivalent to $I_X$ being a complete intersection. For projective toric surfaces, we prove that multiplicity-one generation holds if and only if it holds for every Picard-number-one toric surface dominated by $X$.

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arXiv2026

Mori dream Jacobian elliptic surfaces of Kodaira dimension one

Let $\pi\colon X\to\mathbb P^1$ be a Jacobian elliptic surface over $\mathbb C$, and set $\chi=\chi(\mathcal O_X)\ge3$, so that $\kappa(X)=1$. Assume that the Mordell--Weil group of $\pi$ is finite and that $\pi$ has at least one reducible fiber, the reducible fibers being of types $I_{n_1},\ldots,I_{n_s}$. We prove that the zero section and the components of the reducible fibers generate the closed Mori cone if and only if \[ \sum_{i=1}^s\frac{\lfloor n_i^2/4\rfloor}{n_i}\le\chi. \] If $\sum_i n_i\le2\chi+3$, then $X$ is a Mori dream surface. The proof combines an explicit description of the facets of the cone generated by the curves visible in the fibration with Artin's criterion applied to the null loci of the dual nef rays. We also show that, in Kodaira dimension one, finiteness of both the Mordell--Weil group and the automorphism group does not mply polyhedrality of the Mori cone. In the polyhedral range, we construct a Jacobian elliptic surface with $(\chi,n)=(3,11)$, Picard number $12$, and a big and nef divisor which is not semiample; in particular, this surface is not a Mori dream surface. Finally, for every integer $\rho\ge2$, we construct a Jacobian elliptic surface of Kodaira dimension one and Picard number $\rho$ that is a Mori dream surface.

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arXiv2026

Remarks on hypersurfaces in $\mathbf{P}^1\times Z$

We study the birational geometry of hypersurfaces in projective varieties of the form $\mathbf{P}^1\times Z$, where $Z$ satisfies mild assumptions. Building on recent results of Herrera--Laface--Ugaglia, we study their Cox rings (when finitely generated) in terms of the Cox ring of $Z$. In particular, we obtain a complete picture when $Z$ is a Fano variety of dimension at least 3, and with class group of rank 1.

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arXiv2026

On Cox Rings of Calabi-Yau hypersurfaces

We study the Cox rings of smooth anticanonical Calabi-Yau hypersurfaces in smooth toric Fano varieties. Using the combinatorics of primitive pairs of the ambient Fano polytope and the description of Cox rings of embedded varieties via localizations, we identify several configurations for which the hypersurface is a Mori dream space and obtain explicit presentations of its Cox ring. We also exhibit combinatorial configurations forcing the birational automorphism group to be infinite, yielding in dimensions three and four a dichotomy between finite generation of the Cox ring and infinite birational automorphism group. Finally, for a class of non-Mori dream examples, we prove the Morrison-Kawamata cone conjecture for the movable cone.

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arXiv2025

On K-stability of Fano's last Fanos

We study K-stability of smooth Fano threefolds of Picard rank $2$ and degree $22$ which can be obtained by blowing up a smooth complete intersection of two quadrics in $\mathbb{P}^5$ along a conic. We also describe the automorphism groups of these threefolds.

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arXiv2024

Mori dream singular $K3$ surfaces

We take a first step towards the classification of singular Mori dream $K3$ surfaces. We prove that if the Picard lattice of a singular $K3$ surface is Mori dream, then the surface is Mori dream. Moreover, we show that for singular $K3$ surfaces, of Picard rank two, being Mori dream is equivalent to contain two negative curves intersecting each other, and apply this result to study Mori dreamness of $K3$ surfaces with a singular point of type $A_n$.

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arXiv2024

Equivalent conjectures on blowing-ups of $\mathbb P^2$

We provide a characterization of asymptotical speciality of a nef and big divisor $D$ on an algebraic surface in terms of the arithmetic genus of curves in $D^{\perp}$. As a consequence we prove that the SHGH conjecture for linear systems on the blowing-up $X_r^2$ of the projective plane at points in very general position is equivalent to the fact that each nef class of is non-special. Finally we prove that if $r < 2^n$ then any nef divisor of $X_r^n$ is asymptotically non-special.

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arXiv2024

The Cox ring of an embedded variety

We compute the Cox ring of an embedded variety $X \subseteq Z$ within a Mori dream space, under the assumption that the pullback map induces an isomorphism at the level of divisor class groups. We show that the Cox ring of $X$ is the intersection of finitely many localizations of a quotient image of the Cox ring of $Z$. As a consequence, we provide an algorithm that terminates if and only if the Cox ring of $X$ is finitely generated, thereby generalizing previous works on the subject. We apply these results to compute the Cox ring of hypersurfaces in smooth projective toric varieties.

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arXiv2024

On blowing up minimal toric surfaces

We prove that the Cox ring of the blowing-up of a minimal toric surface of Picard rank two is finitely generated. As part of our proof of this result we provide a necessary and sufficient condition for finite generation of Cox rings of normal projective $\mathbb Q$-factorial surfaces.

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arXiv2022

Effective cone of the blow up of the symmetric product of a curve

Let $C$ be a smooth curve of genus $g \geq 1$ and let $C^{(2)}$ be its second symmetric product. In this note we prove that if $C$ is very general, then the blow-up of $C^{(2)}$ at a very general point has non-polyhedral pseudo-effective cone. The strategy is to consider first the case of hyperelliptic curves and then to show that having polyhedral pseudo-effective cone is a closed property for families of surfaces.

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arXiv2022

Finite generation of Cox rings

In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely generated, the variety admits a quotient presentation by a quasitorus, which resembles the quotient construction of the projective space. We discuss the problem of the finite generation of Cox rings from a geometric perspective and provide examples of both the finitely and non-finitely generated cases.

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arXiv2022

On intrinsic negative curves

Let $\mathbb K$ be an algebraically closed field of characteristic $0$. A curve of $(\mathbb K^*)^2$ arising from a Laurent polynomial in two variables is {\em intrinsic negative} if its tropical compactification has negative self-intersection. The aim of this note is to start a systematic study of these curves and to relate them with the problem of computing Seshadri constants of toric surfaces.

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arXiv2022

Cohen-Macaulay Du Bois singularities with a torus action of complexity one

Using Altmann-Hausen-Suss description of $\mathbb{T}$-varieties via divisorial fans and Kóvacs-Schwede-Smith characterization of Du Bois singularities, we study Cohen-Macaulay Du Bois $\mathbb{T}$-singularities of complexity one. We exhibit cohomological criteria for a $\mathbb{T}$-variety to be Cohen-Macaulay and Du Bois in terms of polyhedral divisors. We give an example of a Cohen-Macaulay Du Bois singularity of complexity one which does not have rational singularities.

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arXiv2022

On linear systems with multiple points on a rational normal curve

We give a closed formula for the dimension of all linear systems in $\mathbb{P}^n$ with assigned multiplicity at arbitrary collections of points lying on a rational normal curve of degree $n$. In particular we give a purely geometric explanation of the speciality of these linear systems, which is due to the presence of certain subvarieties in the base locus: linear spans of points, secant varieties of the rational normal curve or joins between them.

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arXiv2021

Blown-up toric surfaces with non-polyhedral effective cone

We construct examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone, both in characteristic $0$ and in every prime characteristic $p$. As a consequence, we prove that the pseudo-effective cone of the Grothendieck-Knudsen moduli space $\overline M_{0,n}$ of stable rational curves is not polyhedral for $n\geq 10$ in characteristic $0$ and in characteristic $p$, for all primes $p$. Many of these toric surfaces are related to a very interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order. Their analysis in characteristic $p$ relies on tools of arithmetic geometry and Galois representations in the spirit of the Lang-Trotter conjecture, producing toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone in characteristic $0$ and in characteristic $p$, for an infinite set of primes $p$ of positive density.

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arXiv2021

Decomposition algorithms for tensors and polynomials

We give algorithms to compute decompositions of a given polynomial, or more generally mixed tensor, as sum of rank one tensors, and to establish whether such a decomposition is unique. In particular, we present methods to compute the decomposition of a general plane quintic in seven powers, and of a general space cubic in five powers; the two decompositions of a general plane sextic of rank nine, and the five decompositions of a general plane septic. Furthermore, we give Magma implementations of all our algorithms.

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