Related papers: Towards Lang--Vojta via Degeneration

Lasell and Ramachandran show that the existence of rational curves of positive self-intersection on a smooth projective surface $X$ implies that all the finite dimensional linear representations of the fundamental group $\pi_1(X)$ are…

We study the problem of classifying the irreducible projective varieties $X$ of dimension $n\ge 2$ in $\Bbb P^N$ which contain an algebraic family $\Cal F$ of dimension $h+1$ ($h<n$) of subvarieties $Y$ of dimension $n-h$, each one…

Let T be a general bidegree (2,2) divisor in the product of two projective planes. Recently A.Verra proved that the existence of two conic bundle structures (c.b.s.) on T implies a new counterexample to the Torelli theorem for Prym…

Let $V$ be a left vector space over a division ring and let ${\mathcal P}(V)$ be the associated projective space. We describe all finite subsets $X\subset V$ such that every permutation on $X$ can be extended to a linear automorphism of $V$…

We explain how to deduce the degenerate analogue of Ariki's categorification theorem over the ground field C as an application of Schur-Weyl duality for higher levels and the Kazhdan-Lusztig conjecture in finite type A. We also discuss some…

In this paper, we study quasi post-critically finite degenerations for rational maps. We construct limits for such degenerations as geometrically finite rational maps on a finite tree of Riemann spheres. We prove the boundedness for such…

We consider the issue of when the L-polynomial of one curve over $\F_q$ divides the L-polynomial of another curve. We prove a theorem which shows that divisibility follows from a hypothesis that two curves have the same number of points…

We study the noncommutative modular curve (which was already studied by Connes, Manin and Marcolli), and the space of geodesics on the usual modular curve, from the viewpoint of algebraic groups, linear algebra and class field theory. This…

We prove the existence of three irreducible curves $C_{12,m}$ of degree 12 with the same number of cusps and different Alexander polynomials. This exhibits a Zariski triple. Moreover we provide a set of generators for the elliptic threefold…

We prove that a degeneration rationally connected varieties over a field of characteristic zero always contains a geometrically irreducible subvariety which is rationally connected.

Boundary points on the moduli space of pointed curves corresponding to collisions of marked points have modular interpretations as degenerate curves. In this paper, we study degenerations of orbifold projective curves corresponding to…

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group, and B is a block of kG with dihedral defect group D which is Morita equivalent to the principal…

Browning and Vishe used the Hardy-Littlewood circle method to show the moduli space of rational curves on smooth hypersurfaces of low degree is irreducible and of the expected dimension. We reinterpret the circle method geometrically and…

Let $S$ be a finite set of primes. We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of $\mathbb{Z}_{S}$-points on integral models of Hilbert modular varieties, extending a result of D.Helm…

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the…

We study entry loci of varieties and their irreducibility from the perspective of $X$-ranks with respect to a projective variety $X$. These loci are the closures of the points that appear in an $X$-rank decomposition of a general point in…

We prove finiteness of hyperkaehler Lagrangian fibrations in any fixed dimension with fixed Fujiki constant and discriminant of the Beauville-Bogomolov-Fujiki lattice, up to deformation. We also prove finiteness of hyperk\"ahler Lagrangian…

We continue, generalize and expand our study of linear degenerations of flag varieties from [G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, Math. Z. 287 (2017), no. 1-2, 615-654]. We realize partial flag varieties as quiver…

Let $A$ be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic $p$. Suppose $\Delta$ is an infinite set of positive integers, such that…

Given a normal surface singularity $(X, Q)$ and a birational morphism to a non- singular surface $\pi : X \to S$, we investigate the local geometry of the exceptional divisor $L$ of $\pi$. We prove that the dimension of the tangent space to…