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We prove that the abundance conjecture holds on a variety $X$ with mild singularities if $X$ has many reflexive differential forms with coefficients in pluricanonical bundles, assuming the Minimal Model Program in lower dimensions. This…

We construct a surface over $\overline{\mathbb{F}}_p$ with $\pi_1^{\'{e}t}(X) = 1$ that is supersingular -- in the sense that $H^2_{\'{e}t}(X, \mathbb{Q}_{\ell}(1))$ is spanned by algebraic cycles -- but is not unirational. This provides a…

Linked projective spaces are quiver Grassmanians of constant dimension one of certain quiver representations, called linked nets, over special class of quivers, called $\mathbb{Z}^n$-quivers. They were recently introduced as a tool for…

Let $f\in\mathbb{C}[x_1,\ldots,x_s]$ be a weighted homogeneous polynomial having an isolated singularity and $\mathcal{T}_n(f)$ be its higher Nash blowup local algebra. We show that $\mathcal{T}_n(f)$ does not admit negative weighted…

Tyomkin's correspondence theorem states the equality of counts of rational curves of fixed homology class in a toric surface satisfying point and cross-ratio conditions with their tropical counterparts. Such correspondence theorems allow us…

We prove that for a dominant rational self-map $f$ on a quasi-projective variety defined over $\overline{\mathbb{Q}}$, there is a point whose $f$-orbit is well-defined and its arithmetic degree is arbitrarily close to the first dynamical…

We prove that every nontrivial principal $G_m$-bundle over a complete uniformly rational variety is algebraically elliptic in the sense of Gromov.

We consider a finite-dimensional vector space $W\subset K^E$ over an arbitrary field $K$ and an arbitrary set $E$. We show that the set $C(W)\subset 2^E$ consisting of the minimal supports of $W$ are the circuits of a matroid on $E$. In…

We prove general type results for orthogonal modular varieties associated with the moduli of compact hyperk\"ahler manifolds of deformation generalised Kummer type ('deformation generalised Kummer varieties'). In particular, we consider…

We calculate the mixed Hodge numbers of smooth 3-dimensional cluster varieties and show that they are of mixed Tate type. We also study the mixed Hodge structures of the cohomology and intersection cohomology groups of some singular cluster…

This article is a part of a series aimed at classifying normal del Pezzo surfaces of Picard rank one over an algebraically closed field of arbitrary characteristic, up to an isomorphism. The key invariant guiding our classification is the…

We compute the total Chow-Witt rings of the classifying space B\mu_n of the roots of unity, as well as the products BG_m x B\mu_n and B\mu_m x B\mu_n for all m,n greater than or equal to 1 based on the strategy by di Lorenzo and Mantovani…

In this paper, we prove that Bloch's conjecture holds for all smooth, complex, projective surfaces with $p_g=q=0$ and $K^2=9$.

This article is a continuation of my previous paper on miraculous pentagrams published in Annales Scientifiques de Facult\'e des Sciences de Toulouse in 2013. We discuss the moduli space of miraculous pentagrams which is the Del Pezzo…

We define certain stacks of rank one sheaves on a smooth projective variety, and show that they admit proper good moduli spaces. We offer several applications to contractions of subschemes inside Hilbert schemes of points. We construct a…

We survey what is known about Fano threefold weighted complete intersections from the point of view of birational rigidity.

We study quasi-$F^e$-split and quasi-$F$-regular singularities, which generalize Yobuko's quasi-$F$-splitting. We establish Fedder type criteria that characterize these properties for hypersurfaces. These criteria offer explicit tools for…

This is the first article in a series aimed at classifying normal del Pezzo surfaces of Picard rank one over algebraically closed fields of arbitrary characteristic up to an isomorphism. Our guiding invariant is the height of a del Pezzo…

Let $(X,D)$ be an open log del Pezzo surface of rank one, that is, $X$ is a normal projective surface of Picard rank one, the boundary $D$ is a reduced nonzero divisor on $X$, and the anti-log canonical divisor $-(K_X+D)$ is ample. We show…

Let $(X,\Delta)$ be a dlt log Calabi-Yau pair admitting a polarized endomorphism. We show that $(X,\Delta)$ is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety. We provide an example which shows that the…