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We give a new construction of monopole Floer homology for spin-c rational homology 3-spheres. As applications we define two invariants of certain smooth compact 4-manifolds with b_1=1 and b^+=0.

Differential Geometry · Mathematics 2019-12-19 Kim A. Froyshov

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…

Number Theory · Mathematics 2020-06-24 Sho Tanimoto

We give the first examples of nef line bundles on smooth projective varieties over finite fields which are not semi-ample. More concretely, we find smooth curves on smooth projective surfaces over finite fields such that the normal bundle…

Algebraic Geometry · Mathematics 2007-12-14 Burt Totaro

Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and…

Combinatorics · Mathematics 2023-02-20 Anton Ayzenberg

We consider the reduction of Brauer classes on surfaces over number fields, with a view toward applications to rationality and derived equivalence. We show that a Brauer class on a very general polarized K3 surface over a number field…

Algebraic Geometry · Mathematics 2022-03-08 Sarah Frei , Brendan Hassett , Anthony Várilly-Alvarado

We prove that any smooth rational projective surface over the field of complex numbers has an open covering consisting of 3 subsets isomorphic to affine planes.

Algebraic Geometry · Mathematics 2022-03-23 Jorge Caravantes , J. Rafael Sendra , David Sevilla , Carlos Villarino

We discuss the geometry and arithmetic of higher-dimensional analogues of Ch\^atelet surfaces; namely, we describe the structure of their Brauer and Picard groups and show that they can violate the Hasse principle. In addition, we use these…

Number Theory · Mathematics 2012-06-13 Anthony Várilly-Alvarado , Bianca Viray

This is my dissertation about digraphs ordered by pp-constructability. We study in particular smooth digraphs, i.e., digraphs without sources or sinks, tournaments and semicomplete digraphs, orientations of paths and cycles, digraphs with…

Rings and Algebras · Mathematics 2025-01-08 Florian Starke

Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…

Geometric Topology · Mathematics 2016-09-07 Victor A. Vassiliev

Suppose $X$ is a torsor under an abelian variety $A$ over a number field. We show that any adelic point of $X$ that is orthogonal to the algebraic Brauer group of $X$ is orthogonal to the whole Brauer group of $X$. We also show that if…

Number Theory · Mathematics 2018-04-27 Brendan Creutz

This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified…

Algebraic Geometry · Mathematics 2016-08-29 Alena Pirutka

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over Q and F_q(t), and conclude with a…

Algebraic Geometry · Mathematics 2021-07-01 Nicolas Addington , Benjamin Antieau , Sarah Frei , Katrina Honigs

Let $X$ be a smooth projective variety over a finite field $\F$. We discuss the unramified cohomology group $H^3_\nr(X,\Q/\Z(2))$. Several conjectures put together imply that this group is finite. For certain classes of threefolds,…

Algebraic Geometry · Mathematics 2012-07-25 Jean-Louis Colliot-Thélène , Bruno Kahn

Let $X$ be a smooth algebraic curve. Suppose that there exists a triple covering $f : X \to Y$ where $Y$ is a smooth algebraic curve. In this paper, we investigate the existence of morphisms from $X$ to the projective line $\mathbf{P}^1$…

Algebraic Geometry · Mathematics 2008-06-12 Dongsoo Shin

We analyze the Brauer-Manin obstruction to rational points on the K3 surfaces over $\mathbb{Q}$ given by double covers of $\mathbb{P}^2$ ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard…

Algebraic Geometry · Mathematics 2019-10-16 Patrick Corn , Masahiro Nakahara

We construct a sequence of convex polyhedra on n vertices with the property that, as n -> infinity, the fraction of its edge unfoldings that avoid overlap approaches 0, and so the fraction that overlap approaches 1. Nevertheless, each does…

Computational Geometry · Computer Science 2008-01-28 Alex Benton , Joseph O'Rourke

We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into…

Algebraic Geometry · Mathematics 2015-04-06 S. Gorchinskiy , V. Guletskii

Two conjectures relating the Kodaira dimension of a smooth projective variety and existence of number of nowhere vanishing 1-forms on the variety are proposed and verified in dimension 3.

Algebraic Geometry · Mathematics 2007-05-23 Tie Luo , Qi Zhang

We prove that any compact K\"ahler 3-dimensional manifold which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of 'simple manifolds', central in the bimeromorphic…

Algebraic Geometry · Mathematics 2014-01-16 Frédéric Campana , Jean-Pierre Demailly , Misha Verbitsky

We develop a theory of \emph{reduced} Gromov-Witten and stable pair invariants of surfaces and their canonical bundles. We show that classical Severi degrees are special cases of these invariants. This proves a special case of the MNOP…

Algebraic Geometry · Mathematics 2016-05-10 M. Kool , R. P. Thomas