# Algebraic Geometry

## New submissions

[ total of 34 entries: 1-34 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Additive invariants of logarithmic schemes
Subjects: Algebraic Geometry (math.AG)

We lift the decomposition theorems in logarithmic algebraic K-theory obtained by Hagihara and Nizio{\l} to a semi-orthogonal decomposition of the category of perfect complexes over the Kummer flat topos. As a byproduct, we extend Hagihara and Nizio{\l}'s results to a much wider class of log schemes and stacks. Further, we obtain uniform structure theorems which apply, in addition to algebraic K-theory, to all additive invariants of log schemes.

[2]
Title: The 21 reducible polars of Klein's quartic
Comments: 28 pages, contains Singular's script
Subjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC); Combinatorics (math.CO)

We describe the singularities and related properties of the arrangement of 21 reducible polars of Klein's quartic, containing Klein's well-known arrangement of $21$ lines.

[3]
Title: Symplectic invariance of rational surfaces on Kähler manifolds
Subjects: Algebraic Geometry (math.AG)

By work of Koll\'{a}r and Ruan, uniruledness of K\"{a}hler manifolds is an invariant of the underlying symplectic manifold. Zhiyu Tian proved that rational connectedness is a symplectic invariant if the dimension is $\leq 3$. We prove existence of a covering family of rational surfaces assuming positivity of certain gravitational descendants.

[4]
Title: Some Questions in $l-$adic Cohomology
Subjects: Algebraic Geometry (math.AG)

The comparison theorem for a smooth projective variety $X$ over $\mathbb{C}$ tells us that the Betti numbers are independent of $l$. We aim to understand the $l$ independence of Betti numbers for smooth projective varieties $X$ over $k$, where $k$ is an algebraic extension of $\mathbb{F}_p$.

[5]
Title: Constraints on the cohomological correspondence associated to a self map
Authors: K. V. Shuddhodan
Subjects: Algebraic Geometry (math.AG)

In this article we establish some constraints on the eigenvalues for the action of a self map of a proper variety on its $\ell$-adic cohomology. The essential ingredients are a trace formula due to Fujiwara, and the theory of weights.

[6]
Title: Moduli of quiver representations for exceptional collections on surfaces
Subjects: Algebraic Geometry (math.AG)

Suppose $S$ is a smooth projective surface over an algebraically closed field $k$, $\mathcal{L}=\{L_1,\ldots,L_n\}$ is a full strong exceptional collection of line bundles on $S$. Let $Q$ be the quiver associated to this collection. One might hope that $S$ is the moduli space of representation of $Q$ with dimension vector $(1,\ldots,1)$ for a suitably chosen stability condition $\theta$: $S\cong M_\theta$. In this paper, we show that this is the case for many surfaces with such collections.

[7]
Title: Fundamental group of non-singular locus of Lauricella's $F_C$
Subjects: Algebraic Geometry (math.AG)

In this paper, we give a set of generators and relations of the fundamental group of the non-singular locus of Lauricella's hypergeometric functions

[8]
Title: The normal hull and commutator group for nonconnected group schemes
Authors: Giulia Battiston
Subjects: Algebraic Geometry (math.AG); Group Theory (math.GR)

In this short note, we prove that there is a well behaved notion of normal hull for smooth algebraic group schemes over a field and that the commutator group $(G,H)$ is well defined for $H\subset G$ smooth, even when both of them are not connected.

[9]
Title: A short survey on Newton polytopes, tropical geometry and ring of conditions of algebraic torus
Subjects: Algebraic Geometry (math.AG)

The purpose of this note is to give an exposition of some interesting combinatorics and convex geometry concepts that appear in algebraic geometry in relation to counting the number of solutions of a system of polynomial equations in several variables over complex numbers. The exposition is aimed for a general audience in mathematics and we hope to be accessible to undergraduate as well as advance high school students. The topics discussed belong to relatively new, and closely related branches of algebraic geometry which are usually referred to as tropical geometry and toric geometry. These areas make connections between the study of algebra and geometry of polynomials and the combinatorial and convex geometric study of piecewise linear functions. The main results discussed in this note are descriptions of the so-called "ring of conditions" of algebraic torus.

### Cross-lists for Tue, 20 Mar 18

[10]  arXiv:1803.06353 (cross-list from math.RT) [pdf, other]
Title: Potentials for Moduli Spaces of A_m-local Systems on Surfaces
Authors: Efim Abrikosov
Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG)

We study properties of potentials on quivers $Q_{\mathcal{T},m}$ arising from cluster coordinates on moduli spaces of $PGL_{m+1}$-local systems on a topological surface with punctures. To every quiver with potential one can associate a $3d$ Calabi-Yau $A_\infty$-category in such a way that a natural notion of equivalence for quivers with potentials (called "right-equivalence") translates to $A_\infty$-equivalence of associated categories.
For any quiver one can define a notion of a "primitive" potential. Our first result is the description of the space of equivalence classes of primitive potentials on quivers $Q_{\mathcal{T}, m}$. Then we provide a full description of the space of equivalence classes of all \emph{generic} potentials for the case $m = 2$ (corresponds to $PGL_3$-local systems). In particular, we show that it is finite-dimensional. This claim extends results of Gei\ss, Labardini-Fragoso and Schr\"oer who have proved analogous statement in $m=1$ case.
In many cases $3d$ Calabi-Yau $A_\infty$-categories constructed from quivers with potentials are expected to be realized geometrically as Fukaya categories of certain Calabi-Yau $3$-folds. Bridgeland and Smith gave an explicit construction of Fukaya categories for quivers $Q_{\mathcal{T},m=1}$. We propose a candidate for Calabi-Yau $3$-folds that would play analogous role in higher rank cases, $m > 1$. We study their (co)homology and describe a construction of collections of $3$-dimensional spheres that should play a role of generating collections of Lagrangian spheres in corresponding Fukaya categories.

[11]  arXiv:1803.06590 (cross-list from math.RT) [pdf, ps, other]
Title: Cell Decompositions for Rank Two Quiver Grassmannians
Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG); Rings and Algebras (math.RA)

We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cells are in bijection with compatible pairs in a maximal Dyck path as predicted by the theory of cluster algebras. The natural bijection between these two labelings gives a geometric explanation for the appearance of Dyck path combinatorics in the theory of quiver Grassmannians.

[12]  arXiv:1803.06700 (cross-list from math.LO) [pdf, ps, other]
Title: Categoricity of Shimura Varieties
Subjects: Logic (math.LO); Algebraic Geometry (math.AG); Number Theory (math.NT)

We propose a model-theoretic structure for Shimura varieties and give necessary and sufficient conditions to obtain categoricity.

[13]  arXiv:1803.06755 (cross-list from math.AT) [pdf, ps, other]
Title: Witt groups of abelian categories and perverse sheaves
Authors: Jörg Schürmann (Universität Münster), Jon Woolf (University of Liverpool)
Subjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG); K-Theory and Homology (math.KT)

In this paper we study the Witt groups of symmetric and anti-symmetric forms on perverse sheaves on a finite-dimensional topologically stratified space with even dimensional strata. We show that the Witt group has a canonical decomposition as a direct sum of the Witt groups of shifted local systems on strata. We compare this with another splitting decomposition' for Witt classes of perverse sheaves obtained inductively from our main new tool, a splitting relation' which is a generalisation of isotropic reduction.
The Witt groups we study are identified with the (non-trivial) Balmer-Witt groups of the constructible derived category of sheaves on the stratified space, and also with the corresponding cobordism groups defined by Youssin.
Our methods are primarily algebraic and apply more widely. The general context in which we work is that of a triangulated category with duality, equipped with a self-dual t-structure with noetherian heart, glued from self-dual t-structures on a thick subcategory and its quotient.

[14]  arXiv:1803.06901 (cross-list from math.RT) [pdf, ps, other]
Title: Cyclic Sieving and Cluster Duality for Grassmannian
Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Combinatorics (math.CO)

We introduce a decorated configuration space $\mathscr{C}onf_n^\times(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ canonically parametrizes a linear basis of the homogenous coordinate ring of the Grassmannian ${\rm Gr}_a(n)$. We prove that $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ is equivalent to the mirror Landau-Ginzburg model of Grassmannian considered by Marsh-Rietsch and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.

[15]  arXiv:1803.06956 (cross-list from math.AC) [pdf, ps, other]
Title: On a question of Swan
Authors: Dorin Popescu
Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG); K-Theory and Homology (math.KT)

We show that a regular local ring is a filtered inductive limit of regular local rings, essentially of finite type over $\bf Z$.

[16]  arXiv:1803.07017 (cross-list from math.NT) [pdf, ps, other]
Title: A Positive Proportion of Hasse Principle Failures in a Family of Châtelet Surfaces
Authors: Nick Rome
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)

We investigate the family of surfaces defined by the affine equation $$Y^2 + Z^2 = (aT^2 + b)(cT^2 +d)$$ where $\vert ad-bc \vert=1$ and develop asymptotic formulae for the frequency of Hasse principle failures. We show that a positive proportion (roughly 23.7%) of such surfaces fail the Hasse principle, by building on previous work of la Bret\`{e}che and Browning.

### Replacements for Tue, 20 Mar 18

[17]  arXiv:1512.08390 (replaced) [pdf, ps, other]
Title: Dwork families and $\mathcal{D}$-modules
Subjects: Algebraic Geometry (math.AG)
[18]  arXiv:1603.02301 (replaced) [pdf, ps, other]
Title: Constructing Reducible Brill--Noether Curves
Authors: Eric Larson
Subjects: Algebraic Geometry (math.AG)
[19]  arXiv:1605.07543 (replaced) [pdf, other]
Title: On tea, donuts and non-commutative geometry
Authors: Igor Nikolaev
Comments: to appear Surveys in Mathematics and its Applications
Subjects: Algebraic Geometry (math.AG); History and Overview (math.HO); Operator Algebras (math.OA)
[20]  arXiv:1609.02360 (replaced) [pdf, ps, other]
Title: Canonical syzygies of smooth curves on toric surfaces
Subjects: Algebraic Geometry (math.AG); Combinatorics (math.CO)
[21]  arXiv:1611.03103 (replaced) [pdf, ps, other]
Title: An introduction to matrix convex sets and free spectrahedra
Authors: Tom-Lukas Kriel
Comments: 70 pages, an old version of this article was named "Free spectahedra, determinants of monic linear pencils and decomposition of pencils"
Subjects: Algebraic Geometry (math.AG)
[22]  arXiv:1705.01023 (replaced) [pdf, ps, other]
Title: Expectations, Concave Transforms, Chow weights, and Roth's theorem for varieties
Authors: Nathan Grieve
Subjects: Algebraic Geometry (math.AG)
[23]  arXiv:1706.01354 (replaced) [pdf, ps, other]
Title: Non Projected Calabi-Yau Supermanifolds over $\mathbb{P}^2$
Comments: 17 pages. Exposition of the main theorem improved. Typos fixed
Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[24]  arXiv:1709.02006 (replaced) [pdf, ps, other]
Title: Quotients of del Pezzo surfaces of degree 2
Authors: Andrey Trepalin
Comments: Final version, accepted to MMJ. 39 pages, 5 tables
Subjects: Algebraic Geometry (math.AG)
[25]  arXiv:1709.04489 (replaced) [pdf, ps, other]
Title: A remark on the Tate conjecture
Authors: Ben Moonen
Comments: 3 pages; updated version that includes a result in characteristic p. To appear in the J. of Alg. Geom
Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
[26]  arXiv:1711.02752 (replaced) [pdf, ps, other]
Title: Constructing Reducible Brill--Noether Curves II
Authors: Eric Larson
Subjects: Algebraic Geometry (math.AG)
[27]  arXiv:1711.07725 (replaced) [pdf, ps, other]
Title: Effective cycles on the symmetric product of a curve, II: the Abel-Jacobi faces
Comments: 26 pages, 1 figure; v2: strenghtened Theorem A by adding some new Abel-Jacobi facets; improved and clarified some results; fixed a few typos
Subjects: Algebraic Geometry (math.AG)
[28]  arXiv:1801.05682 (replaced) [pdf, ps, other]
Title: Automorphisms of Hilbert schemes of points on a generic projective K3 surface
Authors: Alberto Cattaneo
Comments: 19 pages. v2: Sections 2 and 6 added, with several new major results
Subjects: Algebraic Geometry (math.AG)
[29]  arXiv:1803.03974 (replaced) [pdf, ps, other]
Title: Formality conjecture for K3 surfaces
Subjects: Algebraic Geometry (math.AG)
[30]  arXiv:1602.07500 (replaced) [pdf, ps, other]
Title: Topological comparison theorems for Bredon motivic cohomology
Comments: Corrected indices in main theorem and a few minor changes. To appear, Transactions AMS
Subjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG)
[31]  arXiv:1607.03972 (replaced) [pdf, ps, other]
Title: $F$-singularities under generic linkage
Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
[32]  arXiv:1703.08281 (replaced) [pdf, ps, other]
Title: Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic
Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
[33]  arXiv:1705.08159 (replaced) [pdf, ps, other]
Title: Diagonal forms of higher degree over function fields of $p$-adic curves
Authors: Susanne Pumpluen
Comments: Some small corrections/changes have been done with respect to the first version
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
[34]  arXiv:1802.10012 (replaced) [pdf, ps, other]
Title: Integral points on generalised affine Châtelet surfaces