Related papers: Tangencial base points on algebraic stacks
We study a relation between roots of characteristic polynomials and intersection points of line arrangements. Using these results, we obtain a lot of applications for line arrangements. Namely, we give (i) a generalized addition theorem for…
We describe new explicit examples of moduli spaces of Bridgeland semistable objects on surfaces, parametrizing objects whose numerical class agrees with the class of a point. This follows ideas of Tramel and Xia, using stability conditions…
We compute the fundamental groups of non-singular analytic Deligne-Mumford curves, classify the simply connected ones, and classify analytic Deligne-Mumford curves by their uniformization type. As a result, we find an explicit presentation…
In this paper a generalized topological central point theorem is proved for maps of a simplex to finite-dimensional metric spaces. Similar generalizations of the Tverberg theorem are considered.
In this paper, we describe a general theory of "spaces with structure sheaves." Specializations of this theory include the classical theory of schemes, the theory of Deligne-Mumford stacks, and their derived generalizations.
We study the stable pair theory on toric surfaces and determine the virtual tangent space over the fixed point loci. Further, we present a program to compute the virtual Euler characteristic, illustrated by the case of the projective plane.…
There is a conjecture, that the torsionfreeness of the module of differentials in a point of an algebraic or algebroid curve should imply that the curve is non singular at that point. A report on the main results is given.
After recalling the basic notions concerning Higgs-Grassmannian schemes, I review how these latter can be used to define generalisations of the notion of positivity conditions, such as numerically flatness, which "feel" the Higgs field.…
This is the second paper in a series on intrinsic Donaldson-Thomas theory, a framework for studying the enumerative geometry of general algebraic stacks. In this paper, we present the construction of Donaldson-Thomas invariants for general…
We develop a valuation-theoretic framework for studying tangent cones of torsion-free sheaves on algebraic varieties. To analyze these objects, we introduce a slope stability theory, including the Harder-Narasimhan filtrations, for finitely…
Border bases arise as a canonical generalization of Gr\"obner bases. We provide a polyhedral characterization of all order ideals (and hence border bases) that are supported by a zero-dimensional ideal: order ideals that support a border…
We consider linear systems on toric varieties of any dimension, with invariant base points, giving a characterization of special linear systems. We then make a new conjecture for linear systems on rational surfaces.
Let $\mathbf{k}$ be an algebraically closed field of characteristic $\geq 7$ or zero. Let $\mathcal{A}$ be a tame order of global dimension $2$ over a normal surface $X$ over $\mathbf{k}$ such that…
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…
Let $f:U\to{\mathbb A}^1$ be a regular function on a smooth scheme $U$ over a field $\mathbb K$. Pantev, Toen, Vaquie and Vezzosi (arXiv:1111.3209, arXiv:1109.5213) define the "derived critical locus" Crit$(f)$, an example of a new class of…
In this paper we give a detailed proof of a result we announced a year ago. This result is an effective version of the theorem of Mazur-Kamienny-Merel concerning uniform bounds for rational torsion points on elliptic curves over number…
The Hodge bundle $\omega$ over a modular curve is a square-root of the canonical bundle twisted by the cuspidal divisor, or a theta characteristic, due to the Kodaira--Spencer isomorphism. We prove that, in most cases, a section of a theta…
We characterize transversality, non-transversality properties on the moduli space of genus 0 stable maps to a semipositive symplectic manifold of dimension 4, when GW([point],...,[point]) is enumerative. In particular, we show that the…
In this paper, we study the Heegner points on more general modular curves other than $X_0(N)$, which generalizes Gross' work "Heegner points on $X_0(N)$". The explicit Gross-Zagier formula and the Euler system property are stated in this…
Given a 0-dimensional scheme X in a n-dimensional projective space P^n_K over an arbitrary field K, we use Liaison theory to characterize the Cayley-Bacharach property of X. Our result extends the result for sets of K-rational points given…