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Luca Ugaglia

Publications and source records attributed to Luca Ugaglia.

At least 19 recordsLinked to original sources
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

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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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

Cones of Cycles on blowups of $({\mathbb P}^1)^n$

We study cones of pseudoeffective cycles on the blow up of $({\mathbb P}^1)^n$ at points in very general position, proving some results concerning their structure. In particular we show that in some cases they turn out to be generated by exceptional classes and fiber classes relatively to the projections onto a smaller number of copies of projective lines.

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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

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

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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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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arXiv2020

On base loci of higher fundamental forms of toric varieties

We study the base locus of the higher fundamental forms of a projective toric variety $X$ at a general point. More precisely we consider the closure $X$ of the image of a map $({\mathbb C}^*)^k\to {\mathbb P}^n$, sending $t$ to the vector of Laurent monomials with exponents $p_0,\dots,p_n\in {\mathbb Z}^k$. We prove that the $m$-th fundamental form of such an $X$ at a general point has non empty base locus if and only if the points $p_i$ lie on a suitable degree-$m$ affine hypersurface. We then restrict to the case in which the points $p_i$ are all the lattice points of a lattice polytope and we give some applications of the above result. In particular we provide a classification for the second fundamental forms on toric surfaces, and we also give some new examples of weighted $3$-dimensional projective spaces whose blowing up at a general point is not Mori dream.

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arXiv2019

Cox ring of the generic fiber

Given a surjective morphism $π\colon X\to Y$ of normal varieties satisfying some regularity hypotheses we prove how to recover a Cox ring of the generic fiber of $π$ from the Cox ring of $X$. As a corollary we show that in some cases it is also possible to recover the Cox ring of a very general fiber, and finally we give an application in the case of the blowing-up of a toric fiber space.

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arXiv2015

On del Pezzo elliptic varieties of degree $\leq 4$

\special{html: } Let $Y$ be a del Pezzo variety of degree $d\leq 4$ and dimension $n\geq 3$, let $H$ be an ample class such that $-K_Y=(n-1)H$ and let $Z\subset Y$ be a $0$-dimensional subscheme of length $d$ such that the subsystem of elements of $|H|$ with base locus $Z$ gives a rational morphism $π_Z\colon Y\dashrightarrow{\mathbb P}^{n-1}$. Denote by $π\colon X\to {\mathbb P}^{n-1}$ the elliptic fibration obtained by resolving the indeterminacy locus of $π_Z$. Extending the results of [arXiv:1305.3340] we study the geometry of the variety $X$ and we prove that the Mordell-Weil group of $π$ is finite if and only if the Cox ring of $X$ is finitely generated.

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arXiv2013

On cubic elliptic varieties

Let X->P^(n-1) be an elliptic fibration obtained by resolving the indeterminacy of the projection of a cubic hypersurface Y of P^(n+1) from a line L not contained in Y. We prove that the Mordell-Weil group of the elliptic fibration is finite if and only if the Cox ring of X is finitely generated. We also provide a presentation of the Cox ring of X when it is finitely generated.

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arXiv2006

Elementary (-1)-curves of P^3

In this note we deal with rational curves in P^3 which are images of a line by means of a finite sequence of cubo-cubic Cremona transformations. We prove that these curves can always be obtained applying to the line a sequence of such transformations increasing at each step the degree of the curve. As a corollary we get a result about curves that can give speciality for linear systems of P^3.

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