Related papers: Foliations and Rational Connectedness in Positive …

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…

A symplectic manifold is called symplectic rationally connected if there is a non-zero genus zero Gromov-Witten invariant with two point insertions. It is conjectured that every smooth projective rationally connected variety is symplectic…

This article presents a theory of modules with iterative connection. This theory is a generalisation of the theory of modules with connection in characteristic zero to modules over rings of arbitrary characteristic. We show that these…

We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.

Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is…

In this short note we prove that if X is a separably rationally connected variety over an algebraically closed field of positive characteristic, then H^1(X, O_X)=0.

Rice's theorem shows that nontrivial extensional properties of partial recursive functions are undecidable. For finite weighted Boolean optimization/CSP-style slices, a Rice-style structural analogue holds for tractability classification:…

This paper is a continuation of [G-dS1]. We study foliations of two types on Shimura varieties $S$ in characteristic $p$. The first, which we call "tautological foliations", are defined on Hilbert modular varieties, and lift to…

There exists a dispute in philosophy, going back at least to Leibniz, whether is it possible to view the world as a network of relations and relations between relations with the role of objects, between which these relations hold, entirely…

We extend the concept of orbifold to that of branchfold, in order to allow any cone singularities with rational angles, and show why branchfolds naturally fit in the theory of branched coverings. Then, we obtain a geometric goodness theorem…

Let X be a smooth, complete, toric variety. We study those curves C in X that are contractible, in the sense that there exists an equivariant morphism with connected fibers, with source X, that contracts exactly the irreducible curves that…

We give a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in…

Consider a locally cartesian closed category with an object I and a class of trivial fibrations, which admit sections and are stable under pushforward and retract as arrows. Define the fibrations to be those maps whose Leibniz exponential…

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $X\subseteq\mathbb{P}^n_k$ be a quasi-projective variety that is $F$-rational and $F$-pure. We prove that if $H \subseteq \mathbb{P}^n_k$ is a general hyperplane,…

Causality plays an important role in understanding intelligent behavior, and there is a wealth of literature on mathematical models for causality, most of which is focused on causal graphs. Causal graphs are a powerful tool for a wide range…

We extend Hacon--M\textsuperscript{c}Kernan's rational chain connectedness theorem to the complex analytic setting. As a consequence, we prove that the fibers of any resolution of singularities of complex analytic kawamata log terminal…

This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and…

Given a parameterization $\phi$ of a rational plane curve C, we study some invariants of C via $\phi$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the…

The Hodge conjecture is shown to hold for rationally connected fivefolds, or more generally for fivefolds for which the base of the maximal rationally connected fibration is at most 3 dimensional.

We prove that every curve on a rationally connected variety is algebraically equivalent to a (non-effective) integral sum of rational curves.