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Luca Tasin

Publications and source records attributed to Luca Tasin.

At least 19 recordsLinked to original sources
arXiv2026

The Hassett-Keel program via $\Theta$-stability

We study the intrinsic notion of $\alpha$-stability for curves arising from the Beyond GIT approach to moduli spaces. We show that it recovers Deligne-Mumford stability for $9/11<\alpha\le1$, and more generally that the $\alpha$-semistable locus is contained in the known modular compactification of the Hassett-Keel proram for $\alpha>2/3-\varepsilon$. As applications, we also obtain new $\alpha$-stability results for smooth curves. Our approach combines slope inequalities with a degeneration theorem showing that every Gorenstein curve with a non-nodal singularity can be isotrivially degenerated to a Gorenstein curve with a $\mathbb G_m$-action.

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arXiv2026

Rank one foliations on toroidal varieties

Consider a log canonical pair $(X,B)$ such that there is a Cartier divisor $D$ for which $T_X(-\log B) \otimes \mathcal O(D)$ is locally free and globally generated. Let $\mathcal F$ be a log canonical foliation of rank 1 on $X$. We prove that there exists a divisor $\Gamma$ such that $(X, \Gamma)$ is log canonical and $K_X + \Gamma \sim K_{\mathcal F} + D$. We then apply this result to prove several statements on the birational geometry of rank 1 log canonical foliations on log homogeneous varieties.

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arXiv2026

K-stability of Fano weighted hypersurfaces via plt flags and convex geometry

We develop a framework to study the K-stability of weighted Fano hypersurfaces based on a combination of birational and convex-geometric techniques. As an application, we prove that all quasi-smooth weighted Fano hypersurfaces of index 1 with at most two weights greater than 1 are K-stable. We also construct several examples of K-unstable quasi-smooth weighted Fano hypersurfaces of low indices. To prove these results, we establish lower bounds for stability thresholds using the method of Abban-Zhuang, which reduces the problem to lower-dimensional cases. A key feature of our approach is the use of plt flags that are not necessarily admissible.

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arXiv2025

Forms and Complex Manifolds

We study the intersection form $F_X$ on the second cohomology group $H^2(X, \mathbb{Z})$ of a compact K\"ahler manifold $X$ of dimension $n$. Although the structure of $F_X$ is relatively well understood in dimensions two and three, much less is known for $n \geq 4$. We investigate the fundamental properties of $F_X$ in higher dimensions and discuss several applications to birational geometry. Finally, we present a number of open problems concerning the relationship between birational invariants and topological invariants of K\"ahler manifolds.

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arXiv2025

Effective positivity of Hodge bundles and applications

We prove new boundedness results across different areas of algebraic geometry, stemming from a unifying technical starting point: bounding the integer $q > 0$ such that the $q$-th Hodge bundle becomes (semi-)positive for families of stable varieties. This result allows us to show that for stable families $f: X \to T$ of maximal variation with klt general fiber and relative dimension $n$ there exist the following bounds: 1) a lower bound for the Chow-Mumford volume $\left( \lambda_{CM,f} \right)^{\dim T}$ of the form $\delta^{\dim T}$, where $\delta$ is uniform; 2) a uniform lower bound on $K_{X/T}^{n+1}$, when $T$ is a curve; 3) an upper bound for $|\mathrm{Aut}(f)|$ when $T$ is a curve, depending uniformly linearly on $K_{X/T}^{n+1}$. Additionally, we draw several several consequences on the subspaces of the moduli space of stable varieties parametrizing at least one klt variety, such as the positivity of Hodge bundles and a lower bound on the Chow-Mumford volume in terms of the dimension and the volume of the parametrized varieties (the volume is needed only if working on the coarse moduli space). We also give pair versions of the above results with coefficients varying in a DCC set.

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arXiv2024

Delta invariants of weighted hypersurfaces

We give a lower bound for the delta invariant of the fundamental divisor of a quasi-smooth weighted hypersurface. As a consequence, we prove K-stability of a large class of quasi-smooth Fano hypersurfaces of index 1 and of all smooth Fano weighted hypersurfaces of index 1 and 2. The proofs are based on the Abban--Zhuang method and on the study of linear systems on flags of weighted hypersurfaces.

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arXiv2022

Infinitely many families of Sasaki-Einstein metrics on spheres

We show that there exist infinitely many families of Sasaki-Einstein metrics on every odd-dimensional standard sphere of dimension at least $5$. We also show that the same result is true for all odd-dimensional exotic spheres that bound parallelizable manifolds.

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arXiv2021

On K-stability of Fano weighted hypersurfaces

Let $X \subset \mathbb P(a_0,\ldots,a_n)$ be a quasi-smooth weighted Fano hypersurface of degree $d$ and index $I_X$ such that $a_i |d$ for all $i$, with $a_0 \le \ldots \le a_n$. If $I_X=1$, we show that, under a suitable condition, the $\alpha$-invariant of $X$ is greater than or equal to $\dim X/(\dim X+1)$ and $X$ is K-stable. This can be applied in particular to any $X$ as above such that $\dim X \le 3$. If $X$ is general and $I_X < \dim X$, then we show that $X$ is K-stable. We also give a sufficient condition for the finiteness of automorphism groups of quasi-smooth Fano weighted complete intersections.

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arXiv2021

Slope inequalities for KSB-stable and K-stable families

We prove some higher dimensional generalizations of the slope inequality originally due to G. Xiao, and to M. Cornalba and J. Harris. We give applications to families of KSB-stable and K-stable pairs, as well as to the study of the ample cone of the moduli space of KSB-stable varieties. Our proofs relies on the study of the Harder-Narasimhan filtration, and some generalizations of Castelnuovo's and Noether's inequalities.

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arXiv2019

Chern numbers of uniruled threefolds

In this paper we show that the Chern numbers of a smooth Mori fibre space in dimension three are bounded in terms of the underlying topological manifold. We also generalise a theorem of Cascini and the second named author on the boundedness of Chern numbers of certain threefolds to the case of negative Kodaira dimension.

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arXiv2019

On some modular contractions of the moduli space of stable pointed curves

The aim of this paper is to study some modular contractions of the moduli space of stable pointed curves. These new moduli spaces, which are modular compactifications of the moduli space of smooth pointed curves, are related with the minimal model program for the moduli space of stable pointed curves and have been introduced in a previous work of the authors. We interpret them as log canonical models of adjoints divisors and we then describe the Shokurov decomposition of a region of boundary divisors on the moduli space of stable pointed curves.

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arXiv2018

On the first steps of the minimal model program for the moduli space of stable pointed curves

The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation and we study their geometric properties. As a particular case, we recover the first few Hassett-Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.

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arXiv2018

A note on the fibres of Mori fibre spaces

In this note we consider the problem of determining which Fano manifolds can be realised as fibres of a Mori fibre space. In particular, we study the case of toric varieties, Fano manifolds with high index and some Fano manifolds with high Picard rank.

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arXiv2017

Kaehler structures on spin 6-manifolds

We show that many spin 6-manifolds have the homotopy type but not the homeomorphism type of a Kaehler manifold. Moreover, for given Betti numbers, there are only finitely many deformation types and hence topological types of smooth complex projectve spin threefolds of general type. Finally, on a fixed spin 6-manifold, the Chern numbers take on only finitely many values on all possible Kaehler structures.

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arXiv2017

Effective non-vanishing for Fano weighted complete intersections

We show that Ambro-Kawamata's non-vanishing conjecture holds true for a quasi-smooth WCI X which is Fano or Calabi-Yau, i.e. we prove that, if H is an ample Cartier divisor on X, then |H| is not empty. If X is smooth, we further show that the general element of |H| is smooth. We then verify Ambro-Kawamata's conjecture for any quasi-smooth weighted hypersurface. We also verify Fujita's freeness conjecture for a Gorenstein quasi-smooth weighted hypersurface. For the proofs, we introduce the arithmetic notion of regular pairs and enlighten some interesting connection with the Frobenius coin problem.

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arXiv2016

On the number and boundedness of log minimal models of general type

We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective klt pair (X,D) with $K_X+D$ big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of D and the volume of $K_X+D$. We further show that all n-dimensional projective klt pairs (X,D), such that $K_X+D$ is big and nef of fixed volume and such that the coefficients of D are contained in a given DCC set, form a bounded family. It follows that in any dimension, minimal models of general type and bounded volume form a bounded family.

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arXiv2016

Algebraic structures with unbounded Chern numbers

We determine all Chern numbers of smooth complex projective varieties of dimension at least four which are determined up to finite ambiguity by the underlying smooth manifold. We also give an upper bound on the dimension of the space of linear combinations of Chern numbers with that property and prove its optimality in dimension four.

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