Algebraic Geometry

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New submissions for Tue, 14 May 24

[1]  arXiv:2405.06883 [pdf, ps, other]

Title: Chow stability of $λ$-stable toric varieties

Authors: King leung Lee, Naoto Yotsutani

Comments: 36pages. Comments are welcome!

Subjects: Algebraic Geometry (math.AG)

For a given polarized toric variety, we define the notion of $\lambda$-stability which is a natural generalization of uniform K-stability. At the neighbourhoods of the vertices of the corresponding moment polytope $\Delta$, we consider appropriate triangulations and give a sufficient criteria for a $\lambda$-stable polarized toric variety $(X,L)$ to be asymptotically Chow polystable when the obstruction of asymptotic Chow semistability (the Futaki-Ono invariant) vanishes. As an application, we prove that any K-semistable polarized smooth toric variety $(X,L)$ with the vanishing Futaki-Ono invariant is asymptotically Chow polystable.

[2]  arXiv:2405.06935 [pdf, ps, other]

Title: Non stable rationality of projective approximations for classifying spaces

Authors: Nobuaki Yagita

Comments: 30 pages

Subjects: Algebraic Geometry (math.AG)

We give many examples of non stable rationalities for projective approximations of classifying spaces. Here we use the new invariant by Benoit-Ottem, and also use the classical unramified cohomology..

[3]  arXiv:2405.07092 [pdf, other]

Title: Belyi Function Decompositions for The Icosahedron of Genus 4

Authors: Natalia Amburg, Mariia Kovaleva

Comments: 22 pages, 15 figures

Subjects: Algebraic Geometry (math.AG)

The icosahedron $I_4$ of genus 4 is a dessin d'enfant embedded in Bring's curve $\mathcal{B}$. The dessin $I_4$ is related in some sense to a regular icosahedron $I_0$ embedded in the complex Riemann sphere. In particular, decompositions of Belyi functions $\beta_{I_0}: \mathbb{CP}^1 \rightarrow \mathbb{CP}^1$ and $\beta_{I_4}: \mathcal{B} \rightarrow \mathbb{CP}^1$ for $I_0$ and $I_4$ have the same lattice. The diagram of $\beta_{I_0}$ decompositions is already known. In the present paper we find $\beta_{I_4}$ decompositions. Note that $\beta_{I_0}$ decomposes into rational functions on $\mathbb{C}P^1$, while in case of $\beta_{I_4}$ we deal with maps between different algebraic curves.

[4]  arXiv:2405.07095 [pdf, ps, other]

Title: Étale motives of geometric origin

Authors: Raphaël Ruimy, Swann Tubach

Comments: 15 pages, comments welcome!

Subjects: Algebraic Geometry (math.AG)

Over qcqs finite-dimensional schemes, we prove that \'etale motives of geometric origin can be characterised by a constructibility property which is purely categorical, giving a full answer to the question "Do all constructible \'etale motives come from geometry?" which dates back to Cisinski and D\'eglise's work. We also show that they afford the continuity property and satisfy h-descent and Milnor excision.

[5]  arXiv:2405.07205 [pdf, ps, other]

Title: On Epimorphism and related problems for linear hypersurfaces

Authors: Parnashree Ghosh, Neena Gupta, Ananya Pal

Subjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC)

Linear hypersurfaces over a field $k$ have been playing a central role in the study of some of the challenging problems on affine spaces. Breakthroughs on such problems have occurred by examining two questions on linear polynomials of the form\\ $H:=\alpha(X_1,\dots,X_m)Y - F(X_1,\dots, X_m,Z,T)\in D:=k[X_1,\ldots,X_m, Y,Z,T]$: (i) Whether the affine variety $\mathbb{V}\in \mathbb{A}^{m+3}_k$ defined by $H$ is isomorphic to $\mathbb{A}^{m+2}_k$. (ii) If $\mathbb{V}$ is isomorphic to an affine space, then whether $H$ is a coordinate in $D$. In \cite{adv2}, the first two authors had addressed these questions when $\alpha$ is a monomial of the form $\alpha(X_1,\ldots,X_m) = X_1^{r_1}\dots X_m^{r_m}$; $r_i>1,\, 1 \leqslant i \leqslant m$ and $F$ is of a certain type.
In this paper, using $K$-theory and $\mathbb{G}_a$-actions, we address these questions for a wider family of linear varieties.
In particular, we show that when the characteristic of $k$ is zero, $F \in k[Z,T]$ and $H$ defines a hyperplane (i.e., the affine variety $\mathbb{V}$ defined by $H$ is an affine space), then $H$ is a coordinate in $D$ along with $X_1, X_2, \dots, X_m$. As a consequence we obtain a certain families of higher dimensional linear hyperplanes satisfying the Abhyankar-Sathaye conjecture on the Epimorphism Problem. Our results in arbitrary characteristic yield counter examples to the Zariski Cancellation Problem in positive characteristic.

[6]  arXiv:2405.07225 [pdf, other]

Title: Classification of Dupin Cyclidic Cubes by Their Singularities

Authors: Jean Michel Menjanahary, Eriola Hoxhaj, Rimvydas Krasauskas

Comments: 34 pages, 31 figures

Subjects: Algebraic Geometry (math.AG)

Triple orthogonal coordinate systems having coordinate lines as circles or straight lines are considered. Technically, they are represented by trilinear rational quaternionic maps and are called Dupin cyclidic cubes, naturally generalizing the bilinear rational quaternionic parametrizations of principal patches of Dupin cyclides. Dupin cyclidic cubes and their singularities are studied and classified up to M\"obius equivalency in Euclidean space.

[7]  arXiv:2405.07247 [pdf, ps, other]

Title: Regular nilpotent partial Hessenberg varieties

Authors: Tatsuya Horiguchi

Comments: 28 pages, 3 figures

Subjects: Algebraic Geometry (math.AG); Combinatorics (math.CO)

Regular nilpotent Hessenberg varieties are subvarieties of full flag varieties, while regular nilpotent partial Hessenberg varieties are subvarieties of partial flag varieties. In this manuscript we first provide a summand formula and a product formula for the Poincar\'e polynomial of regular nilpotent partial Hessenberg varieties. It is well-known that there is an isomorphism between the cohomology rings of partial flag varieties and the invariants in the cohomology rings of full flag varieties under an action of a certain Weyl group by Bernstein-Gelfand-Gelfand. We generalize this result to regular nilpotent partial Hessenberg varieties. More concretely, we give an isomorphism between the cohomology rings of regular nilpotent partial Hessenberg varieties and the invariant subrings of the cohomology rings of regular nilpotent Hessenberg varieties under the certain Weyl group action. Furthermore, we provide a description of the cohomology rings for regular nilpotent partial Hessenberg varieties in terms of the invariants in the logarithmic derivation modules of ideal arrangements, which is a genelarization of the result by Abe-Masuda-Murai-Sato with the author.

[8]  arXiv:2405.07322 [pdf, ps, other]

Title: A Gromov-Witten approach to $G$-equivariant birational invariants

Authors: Leonardo F. Cavenaghi, Lino Grama, Ludmil Katzarkov

Comments: 19 pages. Comments are very welcome

Subjects: Algebraic Geometry (math.AG); Mathematical Physics (math-ph); Differential Geometry (math.DG); Symplectic Geometry (math.SG)

In arXiv:2404.19088, we initiated a program linking birational invariants with smooth ones and offering new interpretations of classical invariants, such as the Kervaire-Milnor invariants. Here, we rely on the profound geometric reasoning provided by Lupercio and Uribe in the early 2000s to establish a connection between Chen-Ruan cohomology and the $G$-birational invariants introduced by Kontsevich, Pestun, and Tschinkel in recent pioneering work, along with presenting applications. The final section of this paper explores conjectures and preliminary results regarding gerbes, connections over orbifolds, discrete torsion, and potential approaches via motivic integration and twisted K-theory for $G$-birationality. Combined with the theory of atoms by Katzarkov, Kontsevich, Pantev, and Yu, the proposal in this paper program will lead to a theory of equivariant atoms.

[9]  arXiv:2405.07402 [pdf, ps, other]

Title: The Ceresa period from tropical homology

Authors: Caelan Ritter

Subjects: Algebraic Geometry (math.AG); Combinatorics (math.CO)

Given a finite graph $G$, we define the Ceresa period $\alpha(G)$ as a tool for studying algebraic triviality of the tropical Ceresa cycle introduced by Zharkov. We show that $\alpha(G) = 0$ if and only if $G$ is of hyperelliptic type; then a theorem of Corey implies that having $\alpha(G) = 0$ is a minor-closed condition with forbidden minors $K_4$ and $L_3$.

[10]  arXiv:2405.07716 [pdf, ps, other]

Title: Equivalent conjectures on blowing-ups of $\mathbb P^2$

Authors: Antonio Laface, Luca Ugaglia, Macarena Vilches

Comments: 10 pages

Subjects: Algebraic Geometry (math.AG)

We provide a characterization of asymptotical speciality of a nef and big divisor $D$ on an algebraic surface in terms of the arithmetic genus of curves in $D^{\perp}$. As a consequence we prove that the SHGH conjecture for linear systems on the blowing-up $X_r^2$ of the projective plane at points in very general position is equivalent to the fact that each nef class of is non-special. Finally we prove that if $r < 2^n$ then any nef divisor of $X_r^n$ is asymptotically non-special.

[11]  arXiv:2405.07849 [pdf, ps, other]

Title: Blow-up invariance for Hodge-Witt sheaves with modulus

Authors: Atsushi Shiho

Comments: 22 pages

Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT)

In this paper, we prove the blow-up invariance for Hodge-Witt sheaves with modulus, which is a generalization of a result of Koizumi for Witt sheaves and that of Kelly-Miyazaki and Koizumi for Hodge sheaves. As a consequence, we obtain the representability of Hodge-Witt sheaves with modulus in the category of motives with modulus under the assumption of resolution of singularities.

[12]  arXiv:2405.07936 [pdf, ps, other]

Title: A Perspective on the Foundations of Derived Analytic Geometry

Authors: Oren Ben-Bassat, Jack Kelly, Kobi Kremnizer

Comments: Comments and feedback very welcome; 274 pages

Subjects: Algebraic Geometry (math.AG); Algebraic Topology (math.AT); Complex Variables (math.CV); Number Theory (math.NT)

We show how one can do algebraic geometry with respect to the category of simplicial objects in an exact category. As a biproduct, we get a theory of derived analytic geometry.

[13]  arXiv:2405.07939 [pdf, ps, other]

Title: Compact moduli of Calabi-Yau cones and Sasaki-Einstein spaces

Authors: Yuji Odaka

Comments: 56 pages

Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Commutative Algebra (math.AC); Differential Geometry (math.DG)

We construct proper moduli algebraic spaces of K-polystable $\mathbb{Q}$-Fano cones (a.k.a. Calabi-Yau cones) or equivalently their links i.e., Sasaki-Einstein manifolds with singularities.
As a byproduct, it gives alternative algebraic construction of proper K-moduli of $\mathbb{Q}$-Fano varieties. In contrast to the previous algebraic proof of its properness ([BHLLX, LXZ]), we do not use the $\delta$-invariants ([FO, BJ]) nor the $L^2$-normalized Donaldson-Futaki invariants. We use the local normalized volume of [Li] and the higher $\Theta$-stable reduction instead.

Cross-lists for Tue, 14 May 24

[14]  arXiv:2405.07000 (cross-list from math.AC) [pdf, ps, other]

Title: Multidegrees, families, and integral dependence

Authors: Yairon Cid-Ruiz, Claudia Polini, Bernd Ulrich

Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)

We study the behavior of multidegrees in families and the existence of numerical criteria to detect integral dependence. We show that mixed multiplicities of modules are upper semicontinuous functions when taking fibers and that projective degrees of rational maps are lower semicontinuous under specialization. We investigate various aspects of the polar multiplicities and Segre numbers of an ideal and introduce a new invariant that we call polar-Segre multiplicities. In terms of polar multiplicities and our new invariants, we provide a new integral dependence criterion for certain families of ideals. By giving specific examples, we show that the Segre numbers are the only invariants among the ones we consider that can detect integral dependence. Finally, we generalize the result of Gaffney and Gassler regarding the lexicographic upper semicontinuity of Segre numbers.

[15]  arXiv:2405.07372 (cross-list from math.AT) [pdf, ps, other]

Title: Spaces of non-resultant systems of real bounded multiplicity determined by a toric variety

Authors: Andrzej Kozlowski, Kohhei Yamaguchi

Comments: arXiv admin note: text overlap with arXiv:2105.14601, arXiv:2009.04255

Subjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG)

For any field $\Bbb F$ and positive integers $m,n,d$ with $(m,n)\not= (1,1)$, Farb and Wolfson defined the certain affine varieties ${\rm Poly}^{d,m}_n(\Bbb F)$ as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others. As a natural generalization of this, for each fan $\Sigma$ and $r$-tuple $D=(d_1,\cdots ,d_r)$ of positive integers, the current authors defined spaces ${\rm Poly}^{D,\Sigma}_n(\Bbb F)$, where $r$ is the number of one dimensional cones in $\Sigma$. These spaces can also be regarded as generalizations of the space ${\rm Hol}^*_D(S^2,X_\Sigma)$ of based rational curves from the Riemann sphere $S^2$ to the toric variety $X_\Sigma$ of degree $D$, where $X_\Sigma$ denotes the toric variety (over $\Bbb C$) corresponding to the fan $\Sigma$. In this paper, we define spaces ${\rm Q}^{D,\Sigma}_n(\Bbb F)$ ($\Bbb F=\Bbb R$ or $\Bbb C$) which are real analogues of ${\rm Poly}^{D,\Sigma}_n(\Bbb F)$ and which can be viewed as a generalizations of spaces considered by Arnold, Vassiliev and others in the context of real singularity theory. We prove that homotopy stability holds for these spaces and compute the stability dimensions explicitly.

[16]  arXiv:2405.07786 (cross-list from math.CV) [pdf, ps, other]

Title: Analyticity theorems for parameter-dependent plurisubharmonic functions

Authors: Bojie He

Comments: 29 pages, to appear in Mathematica Scandinavica (2024). All comments are welcome!

Subjects: Complex Variables (math.CV); Algebraic Geometry (math.AG)

In this paper, we first show that a union of upper-level sets associated to fibrewise Lelong numbers of plurisubharmonic functions is in general a pluripolar subset. Then we obtain analyticity theorems for a union of sub-level sets associated to fibrewise complex singularity exponents of some special (quasi-)plurisubharmonic functions. As a corollary, we confirm that, under certain conditions, the logarithmic poles of relative Bergman kernels form an analytic subset when the (quasi-)plurisubharmonic weight function has analytic singularities. In the end, we give counterexamples to show that the aforementioned sets are in general non-analytic even if the plurisubharmonic function is supposed to be continuous.

[17]  arXiv:2405.07873 (cross-list from math.CT) [pdf, ps, other]

Title: A formal categorical approach to the homotopy theory of dg categories

Authors: Yuki Imamura

Comments: 41 pages

Subjects: Category Theory (math.CT); Algebraic Geometry (math.AG); Representation Theory (math.RT)

We introduce a bicategory which refines the localization of the category of dg categories with respect to quasi-equivalences and investigate its properties via formal category theory. Concretely, we first introduce the bicategory of dg categories $\mathsf{DBimod}$ whose Hom categories are the derived categories of dg bimodules and then define the desired bicategory as the sub-bicategory $\mathsf{DBimod}^\text{rqr}$ consisting only of right quasi-representable dg bimodules. The first half of the paper is devoted to the study of adjunctions and equivalences in these bicategories. We then show that the embedding $\mathsf{DBimod}^\text{rqr} \hookrightarrow \mathsf{DBimod}$ is a proarrow equipment in the sense of Richard J. Wood, which is a framework for formal category theory and makes it possible to talk about (weighted) (co)limits in an abstract way. Thus we obtain the notion of homotopical (co)limits in a dg category, including homotopical shifts and cones, by which we obtain a formal categorical characterization of pretriangulated dg categories. As an immediate application we give a conceptual proof of the fact that the pretriangulatedness is preserved under the gluing procedure.

Replacements for Tue, 14 May 24

[18]  arXiv:1509.03443 (replaced) [pdf, ps, other]

Title: A guide to tropical modifications

Authors: Nikita Kalinin

Comments: updated references

Subjects: Algebraic Geometry (math.AG)

[19]  arXiv:2201.04890 (replaced) [pdf, ps, other]

Title: On the group of automorphisms of Horikawa surfaces

Authors: V. Lorenzo

Comments: Some changes following referee's report. Final version to appear in Comptes Rendus Math\'ematique. 12 pages

Subjects: Algebraic Geometry (math.AG)

[20]  arXiv:2203.09778 (replaced) [pdf, ps, other]

Title: The Hodge conjecture for powers of K3 surfaces of Picard number 16

Authors: Mauro Varesco

Comments: 29 pages, to appear on the Michigan Mathematical Journal

Subjects: Algebraic Geometry (math.AG)

[21]  arXiv:2207.09123 (replaced) [pdf, ps, other]

Title: Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D

Authors: Simon Jacques (UL, IECL)

Subjects: Algebraic Geometry (math.AG); Group Theory (math.GR); Representation Theory (math.RT)

[22]  arXiv:2208.14327 (replaced) [pdf, ps, other]

Title: Some interesting birational morphisms of smooth affine quadric $3$-folds

Authors: Cinzia Bisi, Jonathan D. Hauenstein, Tuyen Trung Truong

Comments: 30 pages. A revised version

Journal-ref: Nonlinearity, 2024

Subjects: Algebraic Geometry (math.AG); Complex Variables (math.CV); Dynamical Systems (math.DS); Numerical Analysis (math.NA)

[23]  arXiv:2301.11593 (replaced) [pdf, ps, other]

Title: The Donovan--Wemyss Conjecture via the Derived Auslander--Iyama Correspondence

Authors: Gustavo Jasso, Bernhard Keller, Fernando Muro

Comments: 25 pages. v4: Corrected several minor typos. v3: New title; final version; to appear in the proceedings of the Abel Symposium 2022: Triangulated categories in representation theory and beyond

Subjects: Algebraic Geometry (math.AG); Quantum Algebra (math.QA); Representation Theory (math.RT)

[24]  arXiv:2302.04673 (replaced) [pdf, other]

Title: Embedded $\mathbb{Q}$-desingularization of real Schubert varieties and application to the relative $\mathbb{Q}$-algebraicity problem

Authors: Enrico Savi

Comments: The introduction has been expanded from previous versions and some comments have been added throughout the paper. A new section for acknowledgements has been added at the end of the document. 33 pages, 3 figures

Subjects: Algebraic Geometry (math.AG)

[25]  arXiv:2303.09754 (replaced) [pdf, ps, other]

Title: On the local dimensions of solutions of Brent equations

Authors: Xin Li, Yixin Bao, Liping Zhang

Comments: Any comments are welcome

Subjects: Algebraic Geometry (math.AG)

[26]  arXiv:2306.07161 (replaced) [pdf, ps, other]

Title: Minimal Terracini loci in projective spaces

Authors: Edoardo Ballico, Maria Chiara Brambilla

Comments: 23 pages. Revised version. Title changed

Subjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC)

[27]  arXiv:2307.04288 (replaced) [pdf, ps, other]

Title: Non-Measure Hyperbolicity of K3 Surfaces and Hilbert Schemes of Points on K3 Surfaces

Authors: Gunhee Cho, David R. Morrison

Comments: We received comments from A. Javanpekar, L. Kamenova, Steven Lu, and Mikhail Zaidenberg after putting the preprint on arXiv, so we made a revision taking these into account. Update the consequence on Hilbert Scheme of points on K3 surfaces

Subjects: Algebraic Geometry (math.AG)

[28]  arXiv:2307.15148 (replaced) [pdf, ps, other]

Title: Isotropic and numerical equivalence for Chow groups and Morava K-theories

Authors: Alexander Vishik

Comments: to appear in Inventiones Mathematicae

Subjects: Algebraic Geometry (math.AG); Algebraic Topology (math.AT); K-Theory and Homology (math.KT)

[29]  arXiv:2308.08119 (replaced) [pdf, ps, other]

Title: Discriminant divisors for conic bundles

Authors: Hiromu Tanaka

Comments: 54 pages

Subjects: Algebraic Geometry (math.AG)

[30]  arXiv:2309.02587 (replaced) [pdf, ps, other]

Title: The singular support of an $\ell$-adic sheaf

Authors: Owen Barrett

Comments: 31 pages. Section 3 reorganized. To appear in the Tunisian Journal of Mathematics

Subjects: Algebraic Geometry (math.AG)

[31]  arXiv:2310.10361 (replaced) [pdf, ps, other]

Title: Linear system of hypersurfaces passing through a Galois orbit

Authors: Shamil Asgarli, Dragos Ghioca, Zinovy Reichstein

Comments: 16 pages; enhanced Theorem 1.3 and added Proposition 8.1

Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT)

[32]  arXiv:2311.05116 (replaced) [pdf, other]

Title: Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications

Authors: Yifan Zhang, Joe Kileel

Subjects: Algebraic Geometry (math.AG); Machine Learning (cs.LG); Numerical Analysis (math.NA)

[33]  arXiv:2312.08126 (replaced) [pdf, ps, other]

Title: Towards the classification of semistable fibrations having exactly five singular fibers

Authors: Margarita Castaneda-Salazar, Margarida Mendes Lopes, Alexis Zamora

Comments: Final version to appear in Mediterr. J. Math. Several changes in exposition

Subjects: Algebraic Geometry (math.AG)

[34]  arXiv:2110.06112 (replaced) [pdf, ps, other]

Title: A Murnaghan-Nakayama rule for Grothendieck polynomials of Grassmannian type

Authors: Khanh Nguyen Duc, Dang Tuan Hiep, Tran Ha Son, Do Le Hai Thuy

Comments: 12 pages, 8 figures

Journal-ref: Annals of Combinatorics, Vol. 28 (2024), 155-168

Subjects: Combinatorics (math.CO); Algebraic Geometry (math.AG); K-Theory and Homology (math.KT); Representation Theory (math.RT)

[35]  arXiv:2207.12439 (replaced) [pdf, ps, other]

Title: Equidistribution and independence of Gauss sums

Authors: Antonio Rojas-León

Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)

[36]  arXiv:2401.07114 (replaced) [pdf, other]

Title: Revisiting Sampson Approximations for Geometric Estimation Problems

Authors: Felix Rydell, Angélica Torres, Viktor Larsson

Subjects: Computer Vision and Pattern Recognition (cs.CV); Algebraic Geometry (math.AG)

[37]  arXiv:2402.13206 (replaced) [pdf, ps, other]

Title: Two Formulas for the Number of Lines on Complex Projective Hypersurfaces

Authors: Javier Álvarez-Vizoso

Subjects: Combinatorics (math.CO); Algebraic Geometry (math.AG)

[38]  arXiv:2403.15960 (replaced) [pdf, other]

Title: The smooth Mordell-Weil group and mapping class groups of elliptic surfaces

Authors: Benson Farb, Eduard Looijenga

Comments: 31 pages, 4 figures. References to work of Hacking-Keating added to this version

Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Number Theory (math.NT)

[39]  arXiv:2403.16938 (replaced) [pdf, other]

Title: Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers

Authors: Gaëtan Borot, Nitin Kumar Chidambaram, Giacomo Umer

Comments: 58 pages, 1 figure

Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Algebraic Geometry (math.AG); Combinatorics (math.CO)

[40]  arXiv:2404.19210 (replaced) [pdf, ps, other]

Title: Character Sheaves on Reductive Lie Algebras in Positive Characteristic

Authors: Tong Zhou

Comments: 14 pages, v2: see the footnote to Theorem 4.1.4

Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG)

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• Replacements