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.
{"title":"An intersection-theoretic proof of the Harer–Zagier fomula","authors":"A. Giacchetto, Danilo Lewa'nski, P. Norbury","doi":"10.14231/AG-2023-004","DOIUrl":"https://doi.org/10.14231/AG-2023-004","url":null,"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.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41809699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let p be a prime, k be a p-closed field of characteristic different from p, and 1→ T → G→ F → 1 be an exact sequence of algebraic groups over k, where T is a torus and F is a finite p-group. In this paper, we study the essential dimension ed(G; p) of G at p. R. Lötscher, M. MacDonald, A. Meyer, and the first author showed that min dim(V )− dim(G) 6 ed(G; p) 6 min dim(W )− dim(G) , where V and W range over the p-faithful and p-generically free k-representations of G, respectively. In the special case where G = F , one recovers the formula for ed(F ; p) proved earlier by N. Karpenko and A. Merkurjev. In the case where F = T , one recovers the formula for ed(T ; p) proved earlier by R. Lötscher et al. In both of these cases, the upper and lower bounds on ed(G; p) given above coincide. In general, there is a gap between them. Lötscher et al. conjectured that the upper bound is, in fact, sharp; that is, ed(G; p) = min dim(W )− dim(G), where W ranges over the p-generically free representations. We prove this conjecture in the case where F is diagonalizable.
{"title":"Essential dimension of extensions of finite groups by tori","authors":"Z. Reichstein, F. Scavia","doi":"10.14231/ag-2021-023","DOIUrl":"https://doi.org/10.14231/ag-2021-023","url":null,"abstract":"Let p be a prime, k be a p-closed field of characteristic different from p, and 1→ T → G→ F → 1 be an exact sequence of algebraic groups over k, where T is a torus and F is a finite p-group. In this paper, we study the essential dimension ed(G; p) of G at p. R. Lötscher, M. MacDonald, A. Meyer, and the first author showed that min dim(V )− dim(G) 6 ed(G; p) 6 min dim(W )− dim(G) , where V and W range over the p-faithful and p-generically free k-representations of G, respectively. In the special case where G = F , one recovers the formula for ed(F ; p) proved earlier by N. Karpenko and A. Merkurjev. In the case where F = T , one recovers the formula for ed(T ; p) proved earlier by R. Lötscher et al. In both of these cases, the upper and lower bounds on ed(G; p) given above coincide. In general, there is a gap between them. Lötscher et al. conjectured that the upper bound is, in fact, sharp; that is, ed(G; p) = min dim(W )− dim(G), where W ranges over the p-generically free representations. We prove this conjecture in the case where F is diagonalizable.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45089827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compute the rational Borel-Moore homology groups for affine determinantal varieties in the spaces of general, symmetric, and skew-symmetric matrices, solving a problem suggested by the work of Pragacz and Ratajski. The main ingredient is the relation with Hartshorne's algebraic de Rham homology theory, and the calculation of the singular cohomology of matrix orbits, using the methods of Cartan and Borel. We also establish the degeneration of the v{C}ech-de Rham spectral sequence for determinantal varieties, and compute explicitly the dimensions of de Rham cohomology groups of local cohomology with determinantal support, which are analogues of Lyubeznik numbers first introduced by Switala. Additionally, in the case of general matrices we further determine the Hodge numbers of the singular cohomology of matrix orbits and of the Borel-Moore homology of their closures, based on Saito's theory of mixed Hodge modules.
{"title":"Borel–Moore homology of determinantal varieties","authors":"A. C. LHorincz, Claudiu Raicu","doi":"10.14231/ag-2023-020","DOIUrl":"https://doi.org/10.14231/ag-2023-020","url":null,"abstract":"We compute the rational Borel-Moore homology groups for affine determinantal varieties in the spaces of general, symmetric, and skew-symmetric matrices, solving a problem suggested by the work of Pragacz and Ratajski. The main ingredient is the relation with Hartshorne's algebraic de Rham homology theory, and the calculation of the singular cohomology of matrix orbits, using the methods of Cartan and Borel. We also establish the degeneration of the v{C}ech-de Rham spectral sequence for determinantal varieties, and compute explicitly the dimensions of de Rham cohomology groups of local cohomology with determinantal support, which are analogues of Lyubeznik numbers first introduced by Switala. Additionally, in the case of general matrices we further determine the Hodge numbers of the singular cohomology of matrix orbits and of the Borel-Moore homology of their closures, based on Saito's theory of mixed Hodge modules.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45655866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $E_N$ denote the coarse moduli space of smooth elliptic surfaces over $mathbb{P}^1$ with fundamental invariant $N$. We compute the Chow ring $A^*(E_N)$ for $Ngeq 2$. For each $Ngeq 2$, $A^*(E_N)$ is Gorenstein with socle in codimension $16$, which is surprising in light of the fact that the dimension of $E_N$ is $10N-2$. As an application, we show that the maximal dimension of a complete subvariety of $E_N$ is $16$. When $N=2$, the corresponding elliptic surfaces are K3 surfaces polarized by a hyperbolic lattice $U$. We show that the generators for $A^*(E_2)$ are tautological classes on the moduli space $mathcal{F}_{U}$ of $U$-polarized K3 surfaces, which provides evidence for a conjecture of Oprea and Pandharipande on the tautological rings of moduli spaces of lattice polarized K3 surfaces.
{"title":"The Chow rings of moduli spaces of elliptic surfaces over ${mathbb P}^1$","authors":"Samir Canning, Bochao Kong","doi":"10.14231/ag-2023-016","DOIUrl":"https://doi.org/10.14231/ag-2023-016","url":null,"abstract":"Let $E_N$ denote the coarse moduli space of smooth elliptic surfaces over $mathbb{P}^1$ with fundamental invariant $N$. We compute the Chow ring $A^*(E_N)$ for $Ngeq 2$. For each $Ngeq 2$, $A^*(E_N)$ is Gorenstein with socle in codimension $16$, which is surprising in light of the fact that the dimension of $E_N$ is $10N-2$. As an application, we show that the maximal dimension of a complete subvariety of $E_N$ is $16$. When $N=2$, the corresponding elliptic surfaces are K3 surfaces polarized by a hyperbolic lattice $U$. We show that the generators for $A^*(E_2)$ are tautological classes on the moduli space $mathcal{F}_{U}$ of $U$-polarized K3 surfaces, which provides evidence for a conjecture of Oprea and Pandharipande on the tautological rings of moduli spaces of lattice polarized K3 surfaces.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46181139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider global generation of sheaves of coinvariants on the moduli space of curves given by simple modules over certain vertex operator algebras, extending results for affine VOAs at integrable levels on stable pointed rational curves. Examples where global generation fails, and further evidence of positivity are given.
{"title":"On global generation of vector bundles on the moduli space of curves from representations of n vertex operator algebras","authors":"Chiara Damiolini, A. Gibney","doi":"10.14231/ag-2023-010","DOIUrl":"https://doi.org/10.14231/ag-2023-010","url":null,"abstract":"We consider global generation of sheaves of coinvariants on the moduli space of curves given by simple modules over certain vertex operator algebras, extending results for affine VOAs at integrable levels on stable pointed rational curves. Examples where global generation fails, and further evidence of positivity are given.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45248928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work we give a geometric proof of Hrushovski’s generalization of the LangWeil estimates on the number of points in the intersection of a correspondence with the graph of Frobenius.
{"title":"The Hrushovski–Lang–Weil estimates","authors":"K. V. Shuddhodan, Y. Varshavsky","doi":"10.14231/ag-2022-020","DOIUrl":"https://doi.org/10.14231/ag-2022-020","url":null,"abstract":"In this work we give a geometric proof of Hrushovski’s generalization of the LangWeil estimates on the number of points in the intersection of a correspondence with the graph of Frobenius.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41626041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a new obstruction to lifting smooth proper varieties in characteristic $p>0$ to characteristic $0$. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of 'etale fundamental groups.
{"title":"An obstruction to lifting to characteristic 0","authors":"H. Esnault, V. Srinivas, J. Stix","doi":"10.14231/ag-2023-011","DOIUrl":"https://doi.org/10.14231/ag-2023-011","url":null,"abstract":"We introduce a new obstruction to lifting smooth proper varieties in characteristic $p>0$ to characteristic $0$. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of 'etale fundamental groups.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48714807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In cite{GL21a} we have developed a new approach to $L^{infty}$-a priori estimates for degenerate complex Monge-Amp`ere equations, when the reference form is closed. This simplifying assumption was used to ensure the constancy of the volumes of Monge-Amp`ere measures. We study here the way these volumes stay away from zero and infinity when the reference form is no longer closed. We establish a transcendental version of the Grauert-Riemenschneider conjecture, partially answering conjectures of Demailly-Pu{a}un cite{DP04} and Boucksom-Demailly-Pu{a}un-Peternell cite{BDPP13}. Our approach relies on a fine use of quasi-plurisubharmonic envelopes. The results obtained here will be used in cite{GL21b} for solving degenerate complex Monge-Amp`ere equations on compact Hermitian varieties.
{"title":"Quasi-plurisubharmonic envelopes 2: Bounds on Monge–Ampère volumes","authors":"V. Guedj, C. H. Lu","doi":"10.14231/AG-2022-021","DOIUrl":"https://doi.org/10.14231/AG-2022-021","url":null,"abstract":"In cite{GL21a} we have developed a new approach to $L^{infty}$-a priori estimates for degenerate complex Monge-Amp`ere equations, when the reference form is closed. This simplifying assumption was used to ensure the constancy of the volumes of Monge-Amp`ere measures. We study here the way these volumes stay away from zero and infinity when the reference form is no longer closed. We establish a transcendental version of the Grauert-Riemenschneider conjecture, partially answering conjectures of Demailly-Pu{a}un cite{DP04} and Boucksom-Demailly-Pu{a}un-Peternell cite{BDPP13}. Our approach relies on a fine use of quasi-plurisubharmonic envelopes. The results obtained here will be used in cite{GL21b} for solving degenerate complex Monge-Amp`ere equations on compact Hermitian varieties.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45020220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the maximal number of planes in a complex smooth cubic fourfold in ${mathbb P}^5$ is $405$, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is $357$, realized by the so-called Clebsch--Segre cubic. Altogether, there are but three (up to projective equivalence) cubics with more than $350$ planes.
{"title":"Planes in cubic fourfolds","authors":"A. Degtyarev, I. Itenberg, J. C. Ottem","doi":"10.14231/ag-2023-007","DOIUrl":"https://doi.org/10.14231/ag-2023-007","url":null,"abstract":"We show that the maximal number of planes in a complex smooth cubic fourfold in ${mathbb P}^5$ is $405$, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is $357$, realized by the so-called Clebsch--Segre cubic. Altogether, there are but three (up to projective equivalence) cubics with more than $350$ planes.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44661053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero, and let $mathscr{X}$ be a $K$-analytic space (in the sense of Huber). In this work, we pursue a non-Archimedean characterization of Campana's notion of specialness. We say $mathscr{X}$ is $K$-analytically special if there exists a connected, finite type algebraic group $G/K$, a dense open subset $mathscr{U}subset G^{text{an}}$ with $text{codim}(G^{text{an}}setminus mathscr{U}) geq 2$, and an analytic morphism $mathscr{U} to mathscr{X}$ which is Zariski dense. With this definition, we prove several results which illustrate that this definition correctly captures Campana's notion of specialness in the non-Archimedean setting. These results inspire us to make non-Archimedean counterparts to conjectures of Campana. As preparation for our proofs, we prove auxiliary results concerning the indeterminacy locus of a meromorphic mapping between $K$-analytic spaces, the notion of pseudo-$K$-analytically Brody hyperbolic, and extensions of meromorphic maps from smooth, irreducible $K$-analytic spaces to the analytification of a semi-abelian variety.
{"title":"A non-Archimedean analogue of Campana's notion of specialness","authors":"J. Morrow, Giovanni Rosso","doi":"10.14231/ag-2023-009","DOIUrl":"https://doi.org/10.14231/ag-2023-009","url":null,"abstract":"Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero, and let $mathscr{X}$ be a $K$-analytic space (in the sense of Huber). In this work, we pursue a non-Archimedean characterization of Campana's notion of specialness. We say $mathscr{X}$ is $K$-analytically special if there exists a connected, finite type algebraic group $G/K$, a dense open subset $mathscr{U}subset G^{text{an}}$ with $text{codim}(G^{text{an}}setminus mathscr{U}) geq 2$, and an analytic morphism $mathscr{U} to mathscr{X}$ which is Zariski dense. With this definition, we prove several results which illustrate that this definition correctly captures Campana's notion of specialness in the non-Archimedean setting. These results inspire us to make non-Archimedean counterparts to conjectures of Campana. As preparation for our proofs, we prove auxiliary results concerning the indeterminacy locus of a meromorphic mapping between $K$-analytic spaces, the notion of pseudo-$K$-analytically Brody hyperbolic, and extensions of meromorphic maps from smooth, irreducible $K$-analytic spaces to the analytification of a semi-abelian variety.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46751585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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