45,710 results on '"Algebraic Geometry (math.AG)"'
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2. Smooth polytopes with negative Ehrhart coefficients
- Author
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Castillo, Federico, Liu, Fu, Nill, Benjamin, and Paffenholz, Andreas
- Subjects
Combinatorics (math.CO) ,Algebraic Geometry (math.AG) - Abstract
In this note, we present examples of smooth lattice polytopes in dimensions 3 and higher whose Ehrhart polynomials have some negative coefficients. This answers negatively a question by Bruns. We also discuss Berline-Vergne valuations as a useful tool in proving Ehrhart positivity results and state several open questions.
- Published
- 2017
3. Connectedness of Brill-Noether loci via degenerations
- Author
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Osserman, Brian
- Subjects
Algebraic Geometry (math.AG) - Abstract
We show that limit linear series spaces for chains of curves are reduced. Using new advances in the foundations of limit linear series, we then use degenerations to study the question of connectedness for spaces of linear series with imposed ramification at up to two points. We find that in general, these spaces may not be connected even when they have positive dimension, but we prove a criterion for connectedness which generalizes the theorem previously proved by Fulton and Lazarsfeld in the case without imposed ramification.
- Published
- 2017
4. Limit linear series and ranks of multiplication maps
- Author
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Liu, Fu, Osserman, Brian, Bigas, Montserrat Teixidor i, and Zhang, Naizhen
- Subjects
Algebraic Geometry (math.AG) - Abstract
We develop a new technique for studying ranks of multiplication maps for linear series via limit linear series and degenerations to chains of genus-1 curves. We use this approach to prove a purely elementary criterion for proving cases of the Maximal Rank Conjecture, and then apply the criterion to several ranges of cases, giving a new proof of the case of quadrics, and also treating several families in the case of cubics. Our proofs do not require restrictions on direction of approach, so we recover new information on the locus in the moduli space of curves on which the maximal rank condition fails.
- Published
- 2017
5. Generation and ampleness of coherent sheaves on abelian varieties, with application to Brill-Noether theory
- Author
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Pareschi, Giuseppe
- Subjects
Ampleness ,Settore MAT/03 ,Mathematics - Algebraic Geometry ,Abelian varieties ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
We introduce a variant of global generation for coherent sheaves on abelian varieties which, under certain circumstances, implies ampleness. This extends a criterion of Debarre asserting that a continuously globally generated coherent sheaf on an abelian variety is ample. We apply this to show the ampleness of certain sheaves, which we call naive Fourier-Mukai-Poincar\'e transforms, and to study the structure of GV sheaves. In turn, one of these applications allows to extend the classical existence and connectedness results of Brill-Noether theory to a wider context, e.g. singular curves equipped with a suitable morphism to an abelian variety. Another application is a general inequality of Brill-Noether type involving the Euler characteristic and the homological dimension., Comment: 23 pages. Proposition 3.1.3 added. Other minor changes
- Published
- 2024
6. Lower bounds on the rank and symmetric rank of real tensors
- Author
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Wang, Kexin and Seigal, Anna
- Subjects
Mathematics - Algebraic Geometry ,Computational Mathematics ,Algebra and Number Theory ,15A69, 14N07, 13P15 ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, only one real counterexample was known. We divide the construction into three steps. The first step involves linear spaces of binary tensors. The second step considers a linear space of larger decomposable tensors. The third step is to verify a conjecture that lower bounds the symmetric rank, on a tensor of interest. We use the construction to build an order six real tensor whose real rank and real symmetric rank differ., 26 pages, 3 figures, v2: updated to match published version
- Published
- 2023
7. Topological types of actions on curves
- Author
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Diego Conti, Alessandro Ghigi, and Roberto Pignatelli
- Subjects
Mathematics - Algebraic Geometry ,Computational Mathematics ,Algebra and Number Theory ,Mathematics::Complex Variables ,FOS: Mathematics ,Algebraic Geometry (math.AG) ,14H37 (Primary) 14Q05, 57M60, 14H15, 14J10 (Secondary) - Abstract
We describe an algorithm that constructs a list of all topological types of holomorphic actions of a finite group on a compact Riemann surface $C$ of genus at least $g \geq 2$ with $C/G \cong \mathbb{P}^1$., v2: corrected definition of Hurwitz equivalence using the braid group
- Published
- 2023
8. Openness of splinter loci in prime characteristic
- Author
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Rankeya Datta and Kevin Tucker
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,FOS: Mathematics ,Computer Science::Social and Information Networks ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Algebraic Geometry (math.AG) - Abstract
A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem is whether the splinter property spreads from a point to an open neighborhood of a noetherian scheme. Our paper addresses this problem in prime characteristic, where we show that a locally noetherian scheme that has finite Frobenius or that is locally essentially of finite type over a quasi-excellent local ring has an open splinter locus. In particular, all varieties over fields of positive characteristic have open splinter loci. Intimate connections are established between the openness of splinter loci and $F$-compatible ideals, which are prime characteristic analogues of log canonical centers. We prove the surprising fact that for a large class of noetherian rings with pure (aka universally injective) Frobenius, the splinter condition is detected by the splitting of a single generically \'etale finite extension. We also show that for a noetherian $\textbf{N}$-graded ring over a field, the homogeneous maximal ideal detects the splinter property., Comment: Comments welcome, minor updates in v2
- Published
- 2023
9. Monomial projections of Veronese varieties: New results and conjectures
- Author
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Liena Colarte-Gómez, Rosa M. Miró-Roig, and Lisa Nicklasson
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,FOS: Mathematics ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Algebraic Geometry (math.AG) - Abstract
In this paper, we consider the homogeneous coordinate rings $A(Y_{n,d}) \cong \mathbb{K}[\Omega_{n,d}]$ of monomial projections $Y_{n,d}$ of Veronese varieties parameterized by subsets $\Omega_{n,d}$ of monomials of degree $d$ in $n+1$ variables where: (1) $\Omega_{n,d}$ contains all monomials supported in at most $s$ variables and, (2) $\Omega_{n,d}$ is a set of monomial invariants of a finite diagonal abelian group $G \subset GL(n+1,\mathbb{K})$ of order $d$. Our goal is to study when $\mathbb{K}[\Omega_{n,d}]$ is a quadratic algebra and, if so, when $\mathbb{K}[\Omega_{n,d}]$ is Koszul or G-quadratic. For the family (1), we prove that $\mathbb{K}[\Omega_{n,d}]$ is quadratic when $s \ge \lceil \frac{n+2}{2} \rceil$. For the family (2), we completely characterize when $\mathbb{K}[\Omega_{2,d}]$ is quadratic in terms of the group $G \subset GL(3,\mathbb{K})$, and we prove that $\mathbb{K}[\Omega_{2,d}]$ is quadratic if and only if it is Koszul. We also provide large families of examples where $\mathbb{K}[\Omega_{n,d}]$ is G-quadratic., Comment: To appear in Journal of Algebra
- Published
- 2023
10. Lefschetz properties of some codimension three Artinian Gorenstein algebras
- Author
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Nancy Abdallah, Nasrin Altafi, Anthony Iarrobino, Alexandra Seceleanu, and Joachim Yaméogo
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Mathematics::Rings and Algebras ,FOS: Mathematics ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Algebraic Geometry (math.AG) ,13E10, 13D40, 13H10 - Abstract
Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of $A$, we show that $A$ is almost strong Lefschetz. This quite modest result is nevertheless arguably the most encompassing so far concerning the strong Lefschetz property for graded codimension three AG algebras.
- Published
- 2023
11. Geometric structure of affine Deligne-Lusztig varieties for GL3
- Author
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Ryosuke Shimada
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) - Abstract
In this paper we study the geometric structure of affine Deligne-Lusztig varieties for $GL_3$ and $b$ basic. We completely determine the irreducible components of the affine Deligne-Lusztig variety. In particular, we classify the cases where all of the irreducible components are classical Deligne-Lusztig varieties times finite-dimensional affine spaces. If this is the case, then the irreducible components are pairwise disjoint.
- Published
- 2023
12. The log product formula
- Author
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Herr, Leo
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
We prove a formula expressing the Log Gromov-Witten Invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of $V$ and $W$. This extends results of F. Qu and Y.P. Lee, who introduced this formula analogously to K. Behrend. The proof requires notions of "log normal cone" and "log virtual fundamental class," as well as modified versions of standard intersection-theoretic machinery adapted to Log Geometry., 33 pages. Comments very welcome!
- Published
- 2023
13. The Intrinsic Normal Cone for Artin Stacks
- Author
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Aranha, Dhyan and Pstrągowski, Piotr
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,FOS: Mathematics ,Geometry and Topology ,Algebraic Geometry (math.AG) - Abstract
We extend the construction of the normal cone of a closed embedding of schemes to any locally of finite type morphism of higher Artin stacks and show that in the Deligne-Mumford case our construction recovers the relative intrinsic normal cone of Behrend and Fantechi. We characterize our extension as the unique one satisfying a short list of axioms, and use it to construct the deformation to the normal cone. As an application of our methods, we associate to any morphism of Artin stacks equipped with a choice of a global perfect obstruction theory a relative virtual fundamental class in the Chow group of Kresch., Comment: Minor changes. Added references
- Published
- 2023
14. E-series of character varieties of non-orientable surfaces
- Author
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Letellier, Emmanuel and Rodriguez-Villegas, Fernando
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Representation Theory (math.RT) ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory - Abstract
In this paper we are interested in two kinds of (stacky) character varieties associated to a compact non-orientable surface. (A) We consider the quotient stack of the space of representations of the fundamental group of this surface to GL(n). (B) We choose a set of k-punctures on the surface and a generic k-tuple of semisimple conjugacy classes of GL(n), and we consider the stack of anti-invariant local systems on the orientation cover of the surface with local monodromies around the punctures given by the prescribed conjugacy classes. We compute the number of points of these spaces over finite fields from which we get a formula for their E-series (a certain specialization of the mixed Poincar\'e series). In case (B), we discuss the mixed Poincar\'e series when the surface is the real projective plane and k=1.
- Published
- 2023
15. Triangulations of non-archimedean curves, semi-stable reduction, and ramification
- Author
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Fantini, Lorenzo and Turchetti, Daniele
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,FOS: Mathematics ,Geometry and Topology ,14D10 (primary) and 14G22, 14E22(secondary) ,Algebraic Geometry (math.AG) - Abstract
Let $K$ be a complete discretely valued field with algebraically closed residue field and let $\mathfrak C$ be a smooth projective and geometrically connected algebraic $K$-curve of genus $g$. Assume that $g\geq 2$, so that there exists a minimal finite Galois extension $L$ of $K$ such that $\mathfrak C_L$ admits a semi-stable model. In this paper, we study the extension $L|K$ in terms of the \emph{minimal triangulation} of $C$, a distinguished finite subset of the Berkovich analytification $C$ of $\mathfrak C$. We prove that the least common multiple $d$ of the multiplicities of the points of the minimal triangulation always divides the degree $[L:K]$. Moreover, if $d$ is prime to the residue characteristic of $K$, then we show that $d=[L:K]$, obtaining a new proof of a classical theorem of T. Saito. We then discuss curves with marked points, which allows us to prove analogous results in the case of elliptic curves, whose minimal triangulations we describe in full in the tame case. In the last section, we illustrate through several examples how our results explain the failure of the most natural extensions of Saito's theorem to the wildly ramified case., Comment: Section 5 has been rewritten and its results strengthened. Exposition improved, several details added, typos fixed, and other minor changes. 45 pages, 6 figures, to appear in Annales de l'Institut Fourier
- Published
- 2023
16. Curves of fixed gonality with many rational points
- Author
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Vermeulen, Floris
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Mathematics - Number Theory ,FOS: Mathematics ,Number Theory (math.NT) ,Algebraic Geometry (math.AG) - Abstract
Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$ $\mathbb{F}_q$-rational points. This is the maximal number of rational points possible. This answers a recent conjecture by Faber--Grantham. Our methods are based on curves on toric surfaces and Poonen's work on squarefree values of polynomials., Comment: 12 pages. The proof doesn't work in characteristic 2, so the theorem and proof have been updated
- Published
- 2023
17. Tautological cycles on tropical Jacobians
- Author
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Gross, Andreas and Shokrieh, Farbod
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Algebraic Geometry (math.AG) ,Physics::Atmospheric and Oceanic Physics ,14T05, 14H40, 14H42, 14H51 - Abstract
The classical Poincar\'e formula relates the rational homology classes of tautological cycles on a Jacobian to powers of the class of Riemann theta divisor. We prove a tropical analogue of this formula. Along the way, we prove several foundational results about real tori with integral structures (and, therefore, tropical abelian varieties). For example, we prove a tropical version of the Appell-Humbert theorem. We also study various notions of equivalences between tropical cycles and their relation to one another.
- Published
- 2023
18. Algebraic Spivak’s theorem and applications
- Author
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Annala, Toni
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::K-Theory and Homology ,FOS: Mathematics ,Geometry and Topology ,Mathematics::Algebraic Topology ,Algebraic Geometry (math.AG) - Abstract
We prove an analogue of Lowrey--Sch\"urg's algebraic Spivak's theorem when working over a base ring $A$ that is either a field or a nice enough discrete valuation ring, and after inverting the residual characteristic exponent $e$ in the coefficients. By this result algebraic bordism groups of quasi-projective derived $A$-schemes can be generated by classical cycles, leading to vanishing results for low degree $e$-inverted bordism classes, as well as to the classification of quasi-smooth projective $A$-schemes of low virtual dimension up to $e$-inverted cobordism. As another application, we prove that $e$-inverted bordism classes can be extended from an open subset, leading to the proof of homotopy invariance of $e$-inverted bordism groups for quasi-projective derived $A$-schemes., Comment: 45 pages. Submitted version
- Published
- 2023
19. Irreducible symplectic varieties from moduli spaces of sheaves on K3 and Abelian surfaces
- Author
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Perego, Arvid and Rapagnetta, Antonio
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Mathematics::Category Theory ,FOS: Mathematics ,Geometry and Topology ,Mathematics::Representation Theory ,Mathematics::Symplectic Geometry ,Algebraic Geometry (math.AG) - Abstract
We show that the moduli spaces of sheaves on a projective K3 surface are irreducible symplectic varieties, and that the same holds for the fibers of the Albanese map of moduli spaces of sheaves on an Abelian surface., Comment: 61 pages, major revisions (the title has changed from the previos version, some of the proofs have been entirely reviewed, the main results have been generalized to a slightly more general notion of general polarization), to appear in Algebraic Geometry
- Published
- 2023
20. Sums of even powers of k-regulous functions
- Author
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Juliusz Banecki and Tomasz Kowalczyk
- Subjects
Mathematics - Algebraic Geometry ,General Mathematics ,26C15, 14P99 (Primary) ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
We provide an example of a nonnegative $k$-regulous function on $\mathbb{R}^n$ for $k\geq 1$ and $n \geq 2$ which cannot be written as a sum of squares of $k$-regulous functions. We then obtain lower bounds for Pythagoras numbers $p_{2d}(\mathcal{R}^k(\mathbb{R}^n))$ of $k$-regulous functions on $\mathbb{R}^n$ for $k\geq 1$ and $n\geq 2$. We also prove that the second Pythagoras number of the ring of $0$-regulous functions $\mathcal{R}^0(X)$ on an irreducible $0$-regulous affine variety $X$ is finite and bounded from above by $2^{\dim X}$., Comment: Final version, to appear in Indagationes Mathmaticae
- Published
- 2023
21. Gram spectrahedra of ternary quartics
- Author
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Vill, Julian
- Subjects
Mathematics - Algebraic Geometry ,Computational Mathematics ,Algebra and Number Theory ,Optimization and Control (math.OC) ,FOS: Mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Optimization and Control - Abstract
The Gram spectrahedron of a real form $f\in\mathbb{R}[\underline{x}]_{2d}$ parametrizes all sum of squares representations of $f$. It is a compact, convex, semi-algebraic set, and we study its facial structure in the case of ternary quartics, i.e. $f\in\mathbb{R}[x,y,z]_4$. We show that the Gram spectrahedron of every smooth ternary quartic has faces of dimension 2, and generically none of dimension 1. We complete the proof that the so called Steiner graph of every smooth quartic is isomorphic to $K_4\coprod K_4$. Moreover, we show that the Gram spectrahedron of a generic psd ternary quartic contains points of all ranks in the Pataki interval., 21 pages, 2 figures
- Published
- 2023
22. Collapsing Calabi–Yau fibrations and uniform diameter bounds
- Author
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Li, Yang
- Subjects
Mathematics - Differential Geometry ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Differential Geometry (math.DG) ,FOS: Mathematics ,Geometry and Topology ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry - Abstract
As a sequel to \cite{Licollapsing}, we study Calabi-Yau metrics collapsing along a holomorphic fibration over a Riemann surface. Assuming at worst canonical singular fibres, we prove a uniform diameter bound for all fibres in the suitable rescaling. This has consequences on the geometry around the singular fibres.
- Published
- 2023
23. Log p-divisible groups associated with log 1-motives
- Author
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Matti Würthen and Heer Zhao
- Subjects
Mathematics - Algebraic Geometry ,Mathematics - Number Theory ,Mathematics::K-Theory and Homology ,General Mathematics ,Mathematik ,FOS: Mathematics ,Number Theory (math.NT) ,Algebraic Geometry (math.AG) ,14L05 (primary), 14A21, 14K99, 11G99 (secondary) - Abstract
We first provide a detailed proof of Kato's classification theorem of log $p$-divisible groups over a noetherian henselian local ring. Exploring Kato's idea further, we then define the notion of a standard extension of a classical finite \'etale group scheme (resp. classical \'etale $p$-divisible group) by a classical finite flat group scheme (resp. classical $p$-divisible group) in the category of finite Kummer flat group log schemes (resp. log $p$-divisible groups), with respect to a given chart on the base. These results are then used to prove that log $p$-divisible groups are formally log smooth. We then study the finite Kummer flat group log schemes $T_n(\mathbf{M}):=H^{-1}(\mathbf{M}\otimes_{\mathbb{Z}}^L\mathbb{Z}/n\mathbb{Z})$ (resp. the log $p$-divisible group $\mathbf{M}[p^{\infty}]$) of a log 1-motive $\mathbf{M}$ over an fs log scheme and show that they are \'etale locally standard extensions. Lastly, we give a proof of the Serre-Tate theorem for log abelian varieties with constant degeneration., Comment: Published online at Canadian Journal of Mathematics. Slightly different from the published version in format! 38 pages
- Published
- 2023
24. Toric degenerations of low-degree hypersurfaces
- Author
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Ilten, Nathan and Lautsch, Oscar
- Subjects
Mathematics - Algebraic Geometry ,14M25, 14J70, 13P10 ,General Mathematics ,FOS: Mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Algebraic Geometry (math.AG) - Abstract
We show that a sufficiently general hypersurface of degree $d$ in $n$-dimensional projective space admits a toric Gr\"obner degeneration if and only if $d\leq 2n-1$., Comment: 6 pages; v2 minor revisions. To appear in Canadian Mathematical Bulletin
- Published
- 2023
25. The Poisson spectrum of the symmetric algebra of the Virasoro algebra
- Author
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Petukhov, Alexey V. and Sierra, Susan J.
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,Rings and Algebras (math.RA) ,FOS: Mathematics ,Mathematics - Rings and Algebras ,Algebraic Geometry (math.AG) ,17B68, 17B63, 17B08, 14L99 - Abstract
Let $W = \mathbb{C}[t,t^{-1}]\partial_t$ be the Witt algebra of algebraic vector fields on $\mathbb{C}^\times$ and let $Vir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. In this paper, we study the Poisson ideal structure of the symmetric algebras of $Vir$ and $W$, as well as several related Lie algebras. We classify prime Poisson ideals and Poisson primitive ideals of $S(Vir)$ and $S(W)$. In particular, we show that the only functions in $W^*$ which vanish on a nontrivial Poisson ideal (that is, the only maximal ideals of $S(W)$ with a nontrivial Poisson core) are given by linear combinations of derivatives at a finite set of points; we call such functions local. Given a local function $\chi\in W^*$, we construct the associated Poisson primitive ideal through computing the algebraic symplectic leaf of $\chi$, which gives a notion of coadjoint orbit in our setting. As an application, we prove a structure theorem for subalgebras of $Vir$ of finite codimension and show in particular that any such subalgebra of $Vir$ contains the central element $z$, substantially generalising a result of Ondrus and Wiesner on subalgebras of codimension 1. As a consequence, we deduce that $S(Vir)/(z-\lambda)$ is Poisson simple if and only if $\lambda \neq 0$., Comment: 51 pages; comments welcome. v2: 52 pages; paper rearranged slightly; classification of maximal Poisson ideals of $S(Vir)$ added; statements of some results corrected. v3: final accepted version. To appear in Compositio Mathematica
- Published
- 2023
26. Rank growth of elliptic curves over 𝑁-th root extensions
- Author
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Shnidman, Ari and Weiss, Ariel
- Subjects
Mathematics - Algebraic Geometry ,Mathematics - Number Theory ,Mathematics::Number Theory ,FOS: Mathematics ,Number Theory (math.NT) ,General Medicine ,11G05, (14G05, 14K05, 11S25) ,Algebraic Geometry (math.AG) - Abstract
Fix an elliptic curve $E$ over a number field $F$ and an integer $n$ which is a power of $3$. We study the growth of the Mordell--Weil rank of $E$ after base change to the fields $K_d = F(\sqrt[2n]{d})$. If $E$ admits a $3$-isogeny, then we show that the average ``new rank'' of $E$ over $K_d$, appropriately defined, is bounded as the height of $d$ goes to infinity. When $n = 3$, we moreover show that for many elliptic curves $E/\mathbb{Q}$, there are no new points on $E$ over $\mathbb{Q}(\sqrt[6]d)$, for a positive proportion of integers $d$. This is a horizontal analogue of a well-known result of Cornut and Vatsal. As a corollary, we show that Hilbert's tenth problem has a negative solution over a positive proportion of pure sextic fields $\mathbb{Q}(\sqrt[6]{d})$. The proofs combine our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the ``correlation trick'', which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves., Comment: 24 pages. Revised following referee comments. Section added with application to Hilbert's 10th problem. To appear in Transactions of the AMS. Comments welcome!
- Published
- 2023
27. Free rational curves on low degree hypersurfaces and the circle method
- Author
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Browning, Tim and Sawin, Will
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Mathematics - Number Theory ,14H10 (11D45, 11P55, 14G05, 14J70) ,FOS: Mathematics ,Number Theory (math.NT) ,Algebraic Geometry (math.AG) - Abstract
We use a function field version of the Hardy-Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre's reformulation of the Batyrev-Manin conjecture in terms of slopes with respect to the tangent bundle., Comment: 34 pages
- Published
- 2023
28. The essential p-dimension of the split finite quasi-simple groups of classical Lie type
- Author
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Hannah Knight
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,FOS: Mathematics ,Group Theory (math.GR) ,Representation Theory (math.RT) ,Algebraic Geometry (math.AG) ,Mathematics - Group Theory ,Mathematics - Representation Theory - Abstract
In this paper, we compute the essential $p$-dimension of the split finite quasi-simple groups of classical Lie type at the defining prime, specifically the quasi-simple groups arising from the general linear and special linear groups, the symplectic groups, and the orthogonal groups.
- Published
- 2023
29. Syzygies in Hilbert schemes of complete intersections
- Author
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Giulio Caviglia and Alessio Sammartano
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,FOS: Mathematics ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Algebraic Geometry (math.AG) ,13D02, 13C40, 13F55, 14C05 - Abstract
Let $ e_1, ..., e_c $ be positive integers and let $ Y \subseteq \mathbb{P}^n$ be the monomial complete intersection defined by the vanishing of $x_1^{e_1}, ..., x_c^{e_c}$. In this paper we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points $Hilb^d(Y)$, and discuss applications to the Hilbert scheme of points $Hilb^d(X)$ of arbitrary complete intersections $X \subseteq \mathbb{P}^n$., Final version
- Published
- 2023
30. Minuscule Schubert varieties of exceptional type
- Author
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Sara Angela Filippini, Jacinta Torres, and Jerzy Weyman
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics - Commutative Algebra ,Mathematics::Representation Theory ,Commutative Algebra (math.AC) ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory - Abstract
We study exceptional minuscule Schubert varieties and provide the defining equations of the defining ideals of their intersection with the big open cell. We also provide the resolutions of these ideals and characterize some of them in terms of fundamental examples of ideals in the theory of Gorenstein ideals., Comment: 21 pages, comments welcome
- Published
- 2023
31. On the connected components of Shimura varieties for CM unitary groups in odd variables
- Author
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Yasuhiro Oki
- Subjects
Mathematics::Group Theory ,Mathematics - Algebraic Geometry ,Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics::Number Theory ,FOS: Mathematics ,Number Theory (math.NT) ,14G35, 20G30, 20G25 ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) - Abstract
We study the prime-to-$p$ Hecke action on the projective limit of the sets of connected components of Shimura varieties with fixed parahoric or Bruhat--Tits level at $p$. In particular, we construct infinitely many Shimura varieties for CM unitary groups in odd variables for which the considering actions are not transitive. We prove this result by giving negative examples on the question of Bruhat--Colliot-Th\'el\`ene--Sansuc--Tits or its variant, which is related to the weak approximation on tori over $\mathbb{Q}$., Comment: 20 pages, comments are welcome
- Published
- 2023
32. Littlewood-Richardson coefficient, Springer fibers and the annihilator varieties of induced representations
- Author
-
Zhuohui Zhang
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,FOS: Mathematics ,20G05, 22E47, 22E50, 14M15, 17B08 ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory - Abstract
For $G=GL(n,\mathbb{C})$ and a parabolic subgroup $P=LN$ with a two-block Levi subgroup $L=GL(n_1)\times GL(n_2)$, the space $G\cdot (\mathcal{\mathcal{O}}+\mathfrak{n})$, where $\mathcal{O}$ is a nilpotent orbit of $\mathfrak{l}$, is a union of nilpotent orbits of $\mathfrak{g}$. In the first part of our main theorem, we use the geometric Sakate equivalence to prove that $\mathcal{O'}\subset G\cdot (\mathcal{\mathcal{O}}+\mathfrak{n})$ if and only if some Littlewood-Richardson coefficients do not vanish. The second part of our main theorem describes the geometry of the space $\mathcal{O}\cap\mathfrak{p}$, which is an important space to study for the Whittaker supports and annihilator varieties of representations of $G$.
- Published
- 2023
33. Automorphisms of Rank-One Generated Hyperbolicity Cones and Their Derivative Relaxations
- Author
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Masaru Ito and Bruno F. Lourenço
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,Mathematics - Metric Geometry ,Optimization and Control (math.OC) ,Applied Mathematics ,FOS: Mathematics ,Metric Geometry (math.MG) ,Geometry and Topology ,Mathematics - Optimization and Control ,Algebraic Geometry (math.AG) ,52A20 (Primary) 22F50, 90C25 (Secondary) - Abstract
A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank one, where the rank is computed with respect to the underlying hyperbolic polynomial. This is a natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of the automorphisms of ROG hyperbolicity cones and their derivative relaxations. One of our main results states that the automorphisms of the derivative relaxations are exactly the automorphisms of the original cone fixing a certain direction. As an application, we completely determine the automorphisms of the derivative relaxations of the nonnegative orthant and of the cone of positive semidefinite matrices. More generally, we also prove relations between the automorphisms of a spectral cone and the underlying permutation-invariant set, which might be of independent interest., Comment: 25 pages. Some minor fixes and changes. To appear at the SIAM Journal on Applied Algebra and Geometry
- Published
- 2023
34. Symmetrically Colored Gaussian Graphical Models with Toric Vanishing Ideals
- Author
-
Jane Ivy Coons, Aida Maraj, Pratik Misra, and Miruna-Stefana Sorea
- Subjects
Algebra and Number Theory ,Applied Mathematics ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Mathematics - Algebraic Geometry ,62R01 ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Algebraic Geometry (math.AG) ,ComputingMethodologies_COMPUTERGRAPHICS ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. The model is called RCOP if this coloring is given by the edge and vertex orbits of a subgroup of the automorphism group of the graph. We show that RCOP Gaussian graphical models on block graphs are toric in the space of covariance matrices and we describe Markov bases for them. To this end, we learn more about the combinatorial structure of these models and their connection with Jordan algebras., Comments are very welcome!
- Published
- 2023
35. Irrational pencils and Betti numbers
- Author
-
Nicolás, Francisco, Py, Pierre, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Geometric Topology (math.GT) ,[MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV] ,Group Theory (math.GR) ,General Medicine ,Mathematics::Algebraic Topology ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,Mathematics - Geometric Topology ,Mathematics - Algebraic Geometry ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,FOS: Mathematics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Mathematics - Group Theory ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry - Abstract
We study irrational pencils with isolated critical points on compact aspherical complex manifolds. We prove that if the set of critical points is nonempty, the homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated. This generalizes a result by Dimca, Papadima and Suciu. By considering self-products of the Cartwright-Steger surface, this allows us to build new examples of smooth projective varieties whose fundamental group has a non-finitely generated homology., Comment: 10 pages, 1 figure, v3: this is the final version, accepted in Annales de la Facult\'e des Sciences de Toulouse
- Published
- 2023
36. Tame fundamental groups of pure pairs and Abhyankar’s lemma
- Author
-
Carvajal-Rojas, Javier and Stäbler, Axel
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Mathematics::Number Theory ,13A35, 14B05, 14H30, 14F18 ,FOS: Mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Algebraic Geometry (math.AG) - Abstract
Let $(R,\mathfrak{m}, k)$ be a strictly local normal $k$-domain of positive characteristic and $P$ be a prime divisor on $X=\text{Spec } R$. We study the Galois category of finite covers over $X$ that are at worst tamely ramified over $P$ in the sense of Grothendieck--Murre. Assuming that $(X,P)$ is a purely $F$-regular pair, our main result is that every Galois cover $f \: Y \to X$ in that Galois category satisfies that $\bigl(f^{-1}(P)\bigr)_{\text{red}}$ is a prime divisor. We shall explain why this should be thought as a (partial) generalization of a classical theorem due to S.S.~Abhyankar regarding the \'etale-local structure of tamely ramified covers between normal schemes with respect to a divisor with normal crossings. Additionally, we investigate the formal consequences this result has on the structure of the fundamental group representing the Galois category. We also obtain a characteristic zero analog by reduction to positive characteristics following Bhatt--Gabber--Olsson's methods., Comment: 45 pages, comments are welcome, typos fixed, shortened down, some parts were rewritten to improve exposition, Proposition 4.8 was removed as it was flawed v3: Major revision, to apper in ANT
- Published
- 2023
37. A Hilbert irreducibility theorem for Enriques surfaces
- Author
-
Gvirtz-Chen, Damián and Mezzedimi, Giacomo
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,14J28, 14G05 (Primary) 14J27, 11R45 (Secondary) ,Number Theory (math.NT) ,Algebraic Geometry (math.AG) - Abstract
We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension 0. Bounding the rank of this object, we prove that a conjecture by Campana and Corvaja--Zannier holds for Enriques surfaces, as well as K3 surfaces of Picard rank greater than 6 apart from a finite list of geometric Picard lattices. Concretely, we prove that such surfaces over finitely generated fields of characteristic 0 satisfy the weak Hilbert property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded., 25 pages. Minor corrections. Accepted for publication in Trans. Amer. Math. Soc
- Published
- 2023
38. Exactness and faithfulness of monoidal functors
- Author
-
Kahn, Bruno, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
- Subjects
Mathematics - Algebraic Geometry ,Mathematics (miscellaneous) ,Mathematics::K-Theory and Homology ,Mathematics::Operator Algebras ,Mathematics::Category Theory ,Applied Mathematics ,FOS: Mathematics ,Category Theory (math.CT) ,Mathematics - Category Theory ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Algebraic Geometry (math.AG) ,Mathematics::Algebraic Topology ,Mathematical Physics - Abstract
Inspired by recent work of Peter O'Sullivan (arXiv:2012.15703), we give a condition under which a faithful monoidal functor between abelian $\otimes$-categories is exact.
- Published
- 2023
39. A CATEGORICAL QUANTUM TOROIDAL ACTION ON THE HILBERT SCHEMES
- Author
-
Yu Zhao
- Subjects
Mathematics - Algebraic Geometry ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,General Mathematics ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Algebraic Geometry (math.AG) ,14D20 - Abstract
We categorify the commutation of Nakajima's Heisenberg operators $P_{\pm 1}$ and their infinitely many counterparts in the quantum toroidal algebra $U_{q_1,q_2}(\ddot{gl_1})$ acting on the Grothendieck groups of Hilbert schemes. By combining our result with arxiv:1804.03645 , one obtains a geometric categorical $U_{q_1,q_2}(\ddot{gl_1})$ action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semi-divisorial log terminal singularities., Comment: V2:typos fixed. 29 pages
- Published
- 2023
40. Universal cohomology theories
- Author
-
LUCA BARBIERI VIALE
- Subjects
Cohomology theory ,Motives ,Algebra and Number Theory ,Mathematics - Category Theory ,K-Theory and Homology (math.KT) ,Mathematics::Algebraic Topology ,Settore MAT/02 - Algebra ,Mathematics - Algebraic Geometry ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics - K-Theory and Homology ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Category Theory (math.CT) ,Mathematics - Algebraic Topology ,Representation Theory (math.RT) ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory - Abstract
We furnish any category of a universal (co)homology theory. Universal (co)homologies and universal relative (co)homologies are obtained by showing representability of certain functors and take values in $R$-linear abelian categories of motivic nature, where $R$ is any commutative unitary ring. Universal homology theory on the one point category yields "hieratic" $R$-modules, i.e. the indization of Freyd's free abelian category on $R$. Grothendieck $\partial$-functors and satellite functors are recovered as certain additive relative homologies on an abelian category for which we also show the existence of universal ones., Comment: References added
- Published
- 2023
41. An intersection-theoretic proof of the Harer–Zagier fomula
- Author
-
Giacchetto, Alessandro, Lewański, Danilo, Norbury, Paul, Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Saclay-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Institut des Hautes Etudes Scientifiques (IHES), IHES, Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), and Institut des Hautes Études Scientifiques (IHES)
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,FOS: Mathematics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Geometry and Topology ,Algebraic Geometry (math.AG) ,Mathematical Physics ,14N10 (Primary) 14H10, 14H60, 05A15 (Secondary) - Abstract
We provide an intersection-theoretic formula for the Euler characteristic of the moduli space of smooth curves. This formula reads purely in terms of Hodge integrals and, as a corollary, the standard calculus of tautological classes gives a new short proof of the Harer-Zagier formula. Our result is based on the Gauss-Bonnet formula, and on the observation that a certain parametrisation of the $\Omega$-class - the Chern class of the universal $r$-th root of the twisted log canonical bundle - provides the Chern class of the log tangent bundle to the moduli space of smooth curves. Being $\Omega$-classes by now employed in many enumerative problems, mostly recently found and at times surprisingly different from each other, we dedicate some work to produce an extensive list of their general properties: extending existing ones, finding new ones, and writing down some only known to the experts., Comment: 13 pages
- Published
- 2023
42. The Distribution of the Number of Real Solutions to the Power Flow Equations
- Author
-
Julia Lindberg, Alisha Zachariah, Nigel Boston, and Bernard Lesieutre
- Subjects
Mathematics - Algebraic Geometry ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,Energy Engineering and Power Technology ,Systems and Control (eess.SY) ,Electrical and Electronic Engineering ,Electrical Engineering and Systems Science - Systems and Control ,Algebraic Geometry (math.AG) - Abstract
In this paper we study the distributions of the number of real solutions to the power flow equations over varying electrical parameters. We introduce a new monodromy and parameter homotopy continuation method for quickly finding all solutions to the power flow equations. We apply this method to find distributions of the number of real solutions to the power flow equations and compare these distributions to those of random polynomials. It is observed that while the power flow equations tend to admit many fewer real-valued solutions than a bound on the total number of complex solutions, for low levels of load they tend to admit many more than a corresponding random polynomial. We show that for cycle graphs the number of real solutions can achieve the maximum bound for specific parameter values and for complete graphs with four or more vertices there are susceptance values that give infinitely many real solutions., Comment: 13 pages
- Published
- 2023
43. Hilbert schemes with two Borel-fixed points
- Author
-
Ritvik Ramkumar
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,Mathematics::Operator Algebras ,13D02, 14C05, 14D22, 14J17 ,FOS: Mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Algebraic Geometry (math.AG) - Abstract
We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel-fixed points and determine when the associated Hilbert schemes or their irreducible components are smooth. In particular, we show that the Hilbert scheme is reduced and has at most two irreducible components. By describing the singularities in a neighbourhood of the Borel-fixed points, we prove that the irreducible components are Cohen-Macaulay and normal. We end by giving many examples of Hilbert schemes with three Borel-fixed points., Comment: To appear in Journal of Algebra
- Published
- 2023
44. Mapping stacks and categorical notions of properness
- Author
-
Halpern-Leistner, Daniel and Preygel, Anatoly
- Subjects
Mathematics - Algebraic Geometry ,14A20, 18-XX, 14F05 ,Algebra and Number Theory ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
One fundamental consequence of a scheme $X$ being proper is that the functor classifying maps from $X$ to any other suitably nice scheme or algebraic stack is representable by an algebraic stack. This result has been generalized by replacing $X$ with a proper algebraic stack. We show, however, that it also holds when $X$ is replaced by many examples of algebraic stacks which are not proper, including many global quotient stacks. This leads us to revisit the definition of properness for stacks. We introduce the notion of a formally proper morphism of stacks and study its properties. We develop methods for establishing formal properness in a large class of examples. Along the way, we prove strong h-descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Our main applications are algebraicity results for mapping stacks and the stack of coherent sheaves on a flat and formally proper stack., Comment: 47 pages, complete re-write of first version: definitions simplified; section on PGE removed; strengthened results on reductive group schemes; added comparison between formal properness and other notions of properness
- Published
- 2023
45. SONC optimization and exact nonnegativity certificates via second-order cone programming
- Author
-
Magron, Victor, Wang, Jie, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), ANR-18-ERC2-0004,COPS,Optimisation garantie pour la vérification des systèmes cyber-physiques(2018), ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), European Project: 813211,H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions (Main Programme), H2020-EU.1.3.1. - Fostering new skills by means of excellent initial training of researchers ,10.3030/813211,POEMA(2019), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, and European Project: 813211,H2020,POEMA(2019)
- Subjects
FOS: Computer and information sciences ,Computer Science - Symbolic Computation ,Algebra and Number Theory ,MathematicsofComputing_NUMERICALANALYSIS ,Symbolic Computation (cs.SC) ,sum of binomial squares ,exact nonnegativity certificate ,Mathematics - Algebraic Geometry ,Computational Mathematics ,rounding-projection algorithm ,Optimization and Control (math.OC) ,sum of nonnegative circuit polynomials ,second-order cone programming ,polynomial optimization ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,second-order conerepresentation ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Algebraic Geometry (math.AG) ,Mathematics - Optimization and Control - Abstract
The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits a SOC representation. Based on this, we give a new algorithm for unconstrained polynomial optimization via SOC programming. We also provide a hybrid numeric-symbolic scheme which combines the numerical procedure with a rounding-projection algorithm to obtain exact nonnegativity certificates. Numerical experiments demonstrate the efficiency of our algorithm for polynomials with fairly large degree and number of variables., 29 pages, 7 tables, 6 figures, extended version of the article published in the proceedings of ISSAC 2020. arXiv admin note: text overlap with arXiv:1906.06179
- Published
- 2023
46. Scrollar invariants, syzygies and representations of the symmetric group
- Author
-
Castryck, Wouter, Vermeulen, Floris, and Zhao, Yongqiang
- Subjects
Mathematics - Algebraic Geometry ,Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Number Theory (math.NT) ,Algebraic Geometry (math.AG) - Abstract
We give an explicit minimal graded free resolution, in terms of representations of the symmetric group $S_d$, of a Galois-theoretic configuration of $d$ points in $\mathbb{P}^{d-2}$ that was studied by Bhargava in the context of ring parametrizations. When applied to the generic fiber of a simply branched degree $d$ cover of $\mathbb{P}^1$ by a relatively canonically embedded curve $C$, our construction gives a new interpretation for the splitting types of the syzygy bundles appearing in its relative minimal resolution. Concretely, our work implies that all these splitting types consist of scrollar invariants of resolvent covers, up to a small shift. This vastly generalizes a prior observation due to Casnati, namely that the first syzygy bundle of a degree $4$ cover splits according to the scrollar invariants of its cubic resolvent. Our work also shows that the splitting types of the syzygy bundles, together with the multi-set of scrollar invariants, belong to a much larger class of multi-sets of invariants that can be attached to $C \to \mathbb{P}^1$: one for each irreducible representation of $S_d$, i.e., one for each partition of $d$., 59 pages
- Published
- 2023
47. A unifying approach to tropicalization
- Author
-
Lorscheid, Oliver
- Subjects
Mathematics - Algebraic Geometry ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
In this paper, we introduce ordered blueprints and ordered blue schemes, which serve as a common language for the different approaches to tropicalizations and which enhances tropical varieties with a schematic structure. As an abstract concept, we consider a tropicalization as a moduli problem about extensions of a given valuation $v:k\to T$ between ordered blueprints $k$ and $T$. If $T$ is idempotent, then we show that a generalization of the Giansiracusa bend relation leads to a representing object for the tropicalization, and that it has yet another interpretation in terms of a base change along $v$. We call such a representing object a scheme theoretic tropicalization. This theory recovers and improves other approaches to tropicalizations as we explain with care in the second part of this text. The Berkovich analytification and the Kajiwara-Payne tropicalization appear as rational point sets of a scheme theoretic tropicalization. The same holds true for its generalization by Foster and Ranganathan to higher rank valuations. The scheme theoretic Giansiracusa tropicalization can be recovered from the scheme theoretic tropicalizations in our sense. We obtain an improvement due to the resulting blueprint structure, which is sufficient to remember the Maclagan-Rinc\'on weights. The Macpherson analytification has an interpretation in terms of a scheme theoretic tropicalization, and we give an alternative approach to Macpherson's construction of tropicalizations. The Thuillier analytification and the Ulirsch tropicalization are rational point sets of a scheme theoretic tropicalization. Our approach yields a generalization to any, possibly nontrivial, valuation $v:k\to T$ with idempotent $T$ and enhances the tropicalization with a schematic structure., Comment: 67 pages; changes to previous version: new title, improved exposition and numerous minor changes; to be published in Trans. Amer. Math. Soc
- Published
- 2023
48. K-THEORY OF NON-ARCHIMEDEAN RINGS II
- Author
-
MORITZ KERZ, SHUJI SAITO, and GEORG TAMME
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Commutative Algebra ,Mathematics::K-Theory and Homology ,General Mathematics ,Mathematics - K-Theory and Homology ,FOS: Mathematics ,K-Theory and Homology (math.KT) ,Algebraic Geometry (math.AG) ,Mathematics::Algebraic Topology - Abstract
We study fundamental properties of analytic $K$-theory of Tate rings such as homotopy invariance, Bass fundamental theorem, Milnor excision, and descent for admissible coverings., v1: 16 pages; v2: 18 pages, final version
- Published
- 2023
49. A regular interpolation problem and its applications
- Author
-
Das, Nilkantha
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,14M05, 14R10, 13B25, 32C15 ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
In this article, we prove the following interpolation problem: if the composition of a function and a regular map between affine varieties is a regular function, then there exists a global regular function of the target variety that coincide with the function on the image of the regular map provided the target variety is factorial and the regular map is almost surjective. We also discuss a few applications of the interpolation problem., Comment: 9 pages. To appear in Comm. Algebra
- Published
- 2023
50. On the Brauer group of a generic Godeaux surface
- Author
-
Alexandrou, Theodosis
- Subjects
Mathematics - Algebraic Geometry ,14F22, 14D06 ,General Mathematics ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
Let $X$ be a Godeaux surface over $\mathbb{C}$ and $q_{X}\colon Y\to X$ be its universal cover. We show that the pullback map $q^{*}_{X}\colon Br(X)\to Br(Y)$ is injective if $\rho(Y)=9$. Our arguments rely on a degeneration technique that also applies to other examples., Comment: 21 pages
- Published
- 2023
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