Related papers: Rational points and rational moduli spaces

We propose a general framework that leads to one-dimensional XX and Hubbard models in full generality, based on the decomposition of an arbitrary vector space (possibly infinite dimensional) into a direct sum of two subspaces, the two…

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 compactify the moduli stack of maps from curves to certain quotient stacks $\mathcal{X}=[W/G]$ with a projective good moduli space, extending previous results from quasimap theory. For doing so, we introduce a new birational…

We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If…

Let H be a homology theory for algebraic varieties over a field k. To a complete k-variety X, one naturally attaches an ideal of the coefficient ring H(k). We show that, when X is regular, this ideal depends only on the upper Chow motive of…

In this article we present an algorithm that uses the graded algebra structure of Hilbert modular forms to compute the adelic $q$-expansion of Hilbert modular forms of weight one as the quotient of Hilbert modular forms of higher weight.…

Let $X$ be a smooth projective rational surface, $D\subset X$ an effective anticanonical curve, $\beta$ a curve class on $X$ and $\mathfrak{d}=\sum w_iP_i$ an effective divisor on $D_{\mathrm{sm}}$. We consider the moduli space…

The space of smooth rational curves of degree $d$ in a projective variety $X$ has compactifications by taking closures in the Hilbert scheme, the moduli space of stable sheaves or the moduli space of stable maps respectively. In this paper…

We construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the "tropical motivic…

We introduce a new route to Hilbert space fragmentation in high dimensions leveraging the group-word formalism. We show that taking strongly fragmented models in one dimension and "lifting" to higher dimensions using subsystem symmetries…

Let V be a rank N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a quadratic form with values in a line bundle L and that S^2 V^* \otimes L is ample. Suppose that the maximum rank of the…

Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is…

We study the moduli functor of flat bundles on smooth, possibly non-proper, algebraic variety $X$ (over a field of characteristic zero). For this we introduce the notion of \emph{formal boundary} of $X$, denoted by $\partial X$, which is a…

Refereed version to appear in Michigan Mathematical Journal. A mistake in the last section of the previous version has been corrected. The new title exactly describes the main result obtained. Building on the geometry of cubic surfaces and…

This paper is a sequel to the paper \cite{refGH}. We relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type. Configuration types arise when one classifies the Hilbert functions and…

Let $Y\subseteq \mathbb{R}^n$ be a closed definable subset and $X\subseteq \mathbb{R}^n$ be a smooth manifold. We construct a version of Morse theory for the restriction to $X$ of the Euclidean distance function from $Y$. This is done using…

Let f : X --> X be a dominant rational map of a projective variety defined over a global field, let d_f be the dynamical degree of f, and let h_X be a Weil height on X relative to an ample divisor. We prove that h_X(f^n(P)) << (d_f + e)^n…

The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method builds upon the formulation introduced in Bertalmio et al., J. Comput. Phys., 174 (2001),…

Let $X\subset \mathbb P^n$ be a degree $d$ hypersurface. We prove that $X$ is GIT stable if the minimal exponent $\widetilde \alpha(X)>\frac{n+1}{d}$ and GIT semistable if $\widetilde \alpha(X)=\frac{n+1}{d}$, resolving a question of Laza.…

An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some…