Related papers: Fano foliations with small algebraic ranks
We give a complete classification of smooth, complex projective Fano 4-folds of Picard number 3 having a prime divisor of Picard number 1. They form 28 distinct families, and we compute the main numerical invariants, study the base locus of…
In this paper, we study the relative anti-canonical divisor $-K_{X/Y}$ of an algebraic fiber space $\phi: X \to Y$, and we reveal relations among positivity conditions of $-K_{X/Y}$, certain flatness of direct image sheaves, and variants of…
We prove a complete classification of degree-$2$ foliations on $\mathbb{P}^n$ in any dimension, assuming they are not algebraically integrable. If $\mathcal{F}$ is such a foliation, then either $\mathcal{F}$ is the linear pull-back of a…
We construct first examples of Fano varieties with torsion in their third cohomology group. The examples are constructed as double covers of linear sections of rank loci of symmetric matrices, and can be seen as higher-dimensional analogues…
We explore the birational structure and invariants of a foliated surface $(X, \mathcal F)$ in terms of the adjoint divisor $K_{\mathcal F}+\epsilon K_X$, $0< \epsilon \ll 1$. We then establish a bound on the automorphism group of an adjoint…
Assuming the abundance conjecture in dimension $d$, we establish a non-algebraicity criterion of foliations: any log canonical foliation of rank $\le d$ with $\nu\neq\kappa$ is not algebraically integrable, answering question of…
We study foliations of space forms by complete hypersurfaces, under some mild conditions on its higher order mean curvatures. In particular, in Euclidean space we obtain a Bernstein-type theorem for graphs whose mean and scalar curvature do…
We construct klt projective varieties with ample canonical class and the smallest known volume. We also find exceptional klt Fano varieties with the smallest known anticanonical volume. We conjecture that our examples have the smallest…
Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of $\mathbb Q$-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation…
We study a particular kind of fiber type contractions between complex, projective, smooth varieties f:X->Y, called Fano conic bundles. This means that X is a Fano variety, and every fiber of f is isomorphic to a plane conic. Denoting by…
We give a characterization theorem for non-degenerated plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree $r$ of a non-degenerated foliation as above provides the minimum number,…
To any rooted tree, we associate a sequence of numbers that we call the logarithmic factorials of the tree. This provides a generalization of Bhargava's factorials to a natural combinatorial setting suitable for studying questions around…
We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves…
We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian's alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system associated to the…
In this paper, we prove a positive characteristic analog of Nakayama's inequality on the numerical Kodaira dimension of algebraic fiber spaces when the generic fibers have nef canonical divisors. To this end, we establish variants of Popa…
We introduce a certain birational invariant of a polarized algebraic variety and use that to obtain upper bounds for the counting functions of rational points on algebraic varieties. Using our theorem, we obtain new upper bounds of Manin…
We construct degenerations of Mukai varieties and linear sections thereof to special unobstructed Fano Stanley-Reisner schemes corresponding to convex deltahedra. This can be used to find toric degenerations of rank one index one Fano…
Given $\epsilon>0$, we show that over an algebraically closed field of characteristic $p>5$, the anticanonical volume of a Fano threefold $X$ (with arbitrary singularities) whose anticanonical divisor has Seshadri constant…
We propose a study of the foliations of the projective plane induced by simple derivations of the polynomial ring in two indeterminates over the complex field. These correspond to foliations which have no invariant algebraic curve nor…
We construct $13$ projective $\mathbb{Q}$-factorial Fano toric varieties and show that for any minimal rational elliptic surface $X$ there is one such toric variety $Z_X$ and a divisor class $\delta_X\in {\rm Cl}(Z_X)$ such that the number…