Searcharxiv⌕ Search

arXiv subjects

Joaquín Moraga

Publications and source records attributed to Joaquín Moraga.

At least 55 records · Page 3Linked to original sources
arXiv2021

On a toroidalization for klt singularities

In this article, we prove a toroidalization principle for finite actions on klt singularities. As an application, we prove that the Jordan property for the regional fundamental group of klt singularities can be realized geometrically: by extracting a toric singularity over the klt germ. In the course of the proof, we will prove statements about finite actions on dual complexes and almost fixed points in the fibers of equivariant Fano type morphisms. Furthermore, we will prove that the rank of a fundamental group of the klt singularity is bounded above by its regularity.

math.AG↗
arXiv2021

Minimal log discrepancies of regularity one

In this article, we use the cone of nef curves to study minimal log discrepancies. The first result is an improvement of the nef cone theorem in the case of log Calabi-Yau dlt pairs. Then, we prove that the ascending chain condition for $n$-dimensional minimal log discrepancies of regularity one holds around zero. Furthermore, we show that there exists an upper bound for the minimal log discrepancy of any $n$-dimensional klt singularity of regularity one.

math.AG↗
arXiv2021

On termination of flips and fundamental groups

In this article, we propose a boundedness conjecture for the regional fundamental group of klt singularities. We prove that this boundedness conjecture, the Zariski closedness of the diminished base locus of $K_X$, and an upper bound for minimal log discrepancies, imply the termination of flips with scaling. We prove the boundedness conjecture in the case of toric singularities, quotient singularites, isolated $3$-fold singularities, and exceptional singularities.

math.AG↗
arXiv2021

A geometric characterization of toric singularities

Given a projective contraction $π\colon X\rightarrow Z$ and a log canonical pair $(X, B)$ such that $-(K_X+B)$ is nef over a neighborhood of a closed point $z\in Z$, one can define an invariant, the complexity of $(X, B)$ over $z \in Z$, comparing the dimension of $X$ and the relative Picard number of $X/Z$ with the sum of the coefficients of those components of $B$ intersecting the fibre over $z$. We prove that the complexity of $(X,B)$ over $z\in Z$ is non-negative and that when it is zero then $(X,\lfloor B \rfloor) \rightarrow Z$ is formally isomorphic to a morphism of toric varieties around $z\in Z$. In particular, considering the case when $π$ is the identity morphism, we get a geometric characterization of singularities that are formally isomorphic to toric singularities. This gives a positive answer to a conjecture due to Shokurov.

math.AG↗
arXiv2021

Kawamata log terminal singularities of full rank

We study Kawamata log terminal singularities of full rank, i.e., $n$-dimensional klt singularities containing a large finite abelian group of rank $n$ in its regional fundamental group. The main result of this article is that klt singularities of full rank degenerate to cones over log crepant equivalent toric quotient varieties. To establish the main theorem, we reduce the proof to the study of Fano type varieties with large finite automorphisms of full rank. We prove that such Fano type varieties are log crepant equivalent toric. Furthermore, any such Fano variety of dimension $n$ contains an open affine subset isomorphic to $\mathbb{G}_m^n$. As a first application, we study complements on klt singularities of full rank. As a second application, we study dual complexes of log Calabi-Yau structures on Fano type varieties with large fundamental group of their smooth locus.

math.AG↗
arXiv2020

Maximal log Fano manifolds are generalized Bott towers

We prove that maximal log Fano manifolds are generalized Bott towers. As an application, we prove that in each dimension, there is a unique maximal snc Fano variety satisfying Friedman's d-semistability condition.

math.AG↗
arXiv2020

Strong $(δ,n)$-complements for semi-stable morphisms

We prove boundedness of global strong $(δ,n)$-complements for generalized $ε$-log canonical pairs of Fano type. We also prove some partial results towards boundedness of local strong $(δ,n)$-complements for semi-stable morphisms. As applications, we prove an effective generalized canonical bundle formula for generalized klt pairs and an effective generalized adjunction formula for exceptional generalized log canonical centers. Moreover, we prove that the existence of strong $(δ,n)$-complements implies a conjecture due to M$^{\rm c}$Kernan concerning the singularities of the base of a Mori fiber space.

math.AG↗
arXiv2020

Small quotient minimal log discrepancies

We prove that for each positive integer $n$ there exists a positive number $ε_n$ so that $n$-dimensional toric quotient singularities satisfy the ACC for mld's on the interval $(0,ε_n)$. In the course of the proof, we will show a geometric Jordan property for finite automorphism groups of affine toric varieties.

math.AG↗
arXiv2020

On weak Zariski decompositions and termination of flips

We prove that termination of lower dimensional flips for generalized klt pairs implies termination of flips for log canonical generalized pairs with a weak Zariski decomposition. Moreover, we prove that the existence of weak Zariski decompositions for pseudo-effective generalized klt pairs implies the existence of minimal models for such pairs.

math.AG↗
arXiv2019

Extracting non-canonical places

Let $(X,B)$ be a log canonical pair and $\mathcal{V}$ be a finite set of divisorial valuations with log discrepancy in $[0,1)$. We prove that there exists a projective birational morphism $π\colon Y\rightarrow X$ so that the exceptional divisors are $\mathbb{Q}$-Cartier and correspond to elements of $\mathcal{V}$. We study how two such models are related. Moreover, we provide an application to the study of deformations of log canonical singularities.

math.AG↗
arXiv2019

Regularity of structure sheaves of varieties with isolated singularities

Let $X\subseteq \mathbb{P}^N$ be a non-degenerate normal projective variety of codimension $e$ and degree $d$ with isolated $\mathbb{Q}$-Gorenstein singularities. We prove that the Castelnuovo-Mumford regularity $\text{reg}(\mathcal{O}_{X})\le d-e$, as predicted by the Eisenbud-Goto regularity conjecture. Such a bound fails for general projective varieties. The main techniques are Noma's classification of non-degenerated projective varieties and Nadel vanishing for multiplier ideals. We also classify the extremal and the next to extremal cases.

math.AG↗
arXiv2019

Bounded deformations of $(ε,δ)$-log canonical singularities

In this paper we study $(ε,δ)$-lc singularites, i.e. $ε$-lc singularities admitting a $δ$-plt blow-up. We prove that $n$-dimensional $(ε,δ)$-lc singularities are bounded up to a deformation, and $2$-dimensional $(ε,δ)$-lc singularities form a bounded family. Furthermore, we give an example which shows that $(ε,δ)$-lc singularities are not bounded in higher dimensions, even in the analytic sense.

math.AG↗
arXiv2018

A boundedness theorem for cone singularities

A cone singularity is a normal affine variety $X$ with an effective one-dimensional torus action with a unique fixed point $x\in X$ which lies in the closure of any orbit of the $k^*$-action. In this article, we prove a boundedness theorem for cone singularities in terms of their dimension, singularities, and isotropies. Given $d$ and $N$ two positive integers and $ε$ a positive real number, we prove that the class of $d$-dimensional $ε$-log canonical cone singularities with isotropies bounded by $N$ forms a bounded family.

math.AG↗
arXiv2018

On minimal log discrepancies and Kollár components

In this article we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$-dimensional $a$-log canonical singularities, with standard coefficients, which admit an $ε$-plt blow-up have minimal log discrepancies belonging to a finite set which only depends on $d,a$ and $ε$. This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

math.AG↗
arXiv2018

Fundamental group of log terminal T-varieties

In this article, we introduce an approach to study the fundamental group of a log terminal T-variety. As applications, we prove the simply connectedness of the spectrum of the Cox ring of a complex Fano variety, we compute the fundamental group of a rational log terminal T-varieties of complexity one, and we study the local fundamental group of log terminal T-singularities with a good torus action and trivial GIT decomposition.

math.AG↗
arXiv2018

On a notion of toric special linear systems

In this note we study linear systems on complete toric varieties $X$ with an invariant point, whose orbit under the action of the automorphism group of $X$ contains the dense torus $T$ of $X$. We give a characterization of such varieties in terms of its defining fan and introduce a new definition of expected dimension of linear systems which consider the contribution given by certain toric subvarieties. Finally, we study degenerations of linear systems on these toric varieties induced by toric degenerations.

math.AG↗
arXiv2018

Bounding singular surfaces via Chern numbers

We prove the existence of a bound on the number of steps of the minimal model program for singular surfaces in terms of discrepancies and top Chern numbers. As an application, we prove that given $R\in\mathbb{R}$ and $ε\in (0,1)$, the class $\mathcal{F}(R,ε)$ of $2$-dimensional pairs $(X,D)$ of general type with $ε$-klt singularities, $D$ with standard coefficients, and $4c_2(X,D)-c_1^2(X,D)\leq R$, forms a bounded family.

math.AG↗