Pub Date : 2020-06-03DOI: 10.1142/s0129167x20500834
C. Shramov
We classify finite groups acting by birational transformations of a non-trivial Severi--Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field $K$ of characteristic zero that has no $K$-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.
{"title":"Automorphisms of cubic surfaces without points","authors":"C. Shramov","doi":"10.1142/s0129167x20500834","DOIUrl":"https://doi.org/10.1142/s0129167x20500834","url":null,"abstract":"We classify finite groups acting by birational transformations of a non-trivial Severi--Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field $K$ of characteristic zero that has no $K$-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116667211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend the so called Chen-Jiang decomposition for pushforward of pluricanocanical bundles to abelian varieties to the setting of klt pairs. We also provide a geometric application of this decomposition.
{"title":"M-regular Decompositions for Pushforwards of Pluricanonical Bundles of Pairs to Abelian Varieties","authors":"Z. Jiang","doi":"10.1093/IMRN/RNAA366","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA366","url":null,"abstract":"We extend the so called Chen-Jiang decomposition for pushforward of pluricanocanical bundles to abelian varieties to the setting of klt pairs. We also provide a geometric application of this decomposition.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123405500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a regular scheme and its reduced closed subscheme, the latter being of finite type over a perfect field of positive characteristic, we define its cotangent bundle restricted to the closed subscheme as a family of vector bundles on smooth schemes over the field endowed with morphisms to the closed subscheme factoring through the Frobenius. For a constructible complex on the etale site of the scheme, we introduce the condition to be micro-supported on a closed conical subset in the cotangent bundle. We compute the singular supports of certain Kummer sheaves of rank 1.
{"title":"Cotangent bundles and micro-supports in mixed characteristic case","authors":"Takeshi Saito","doi":"10.2140/ant.2022.16.335","DOIUrl":"https://doi.org/10.2140/ant.2022.16.335","url":null,"abstract":"For a regular scheme and its reduced closed subscheme, the latter being of finite type over a perfect field of positive characteristic, we define its cotangent bundle restricted to the closed subscheme as a family of vector bundles on smooth schemes over the field endowed with morphisms to the closed subscheme factoring through the Frobenius. For a constructible complex on the etale site of the scheme, we introduce the condition to be micro-supported on a closed conical subset in the cotangent bundle. We compute the singular supports of certain Kummer sheaves of rank 1.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116722378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the Heller relative $K_{0}$ group of the map $mathbb{P}_{X}^{r} to mathbb{P}_{S}^{r},$ where $X$ and $S$ are quasi-projective schemes over a commutative ring. More precisely, we prove that the projective bundle formula holds for Heller's relative $K_{0},$ provided $X$ is flat over $S.$ As a corollary, we get a description of the relative group $K_{0}(mathbb{P}_{X}^{r} to mathbb{P}_{S}^{r})$ in terms of generators and relations, provided $X$ is affine and flat over $S.$
{"title":"Projective bundle formula for Heller's relative $K_{0}$","authors":"V. Sadhu","doi":"10.2996/kmj/1605063630","DOIUrl":"https://doi.org/10.2996/kmj/1605063630","url":null,"abstract":"In this article, we study the Heller relative $K_{0}$ group of the map $mathbb{P}_{X}^{r} to mathbb{P}_{S}^{r},$ where $X$ and $S$ are quasi-projective schemes over a commutative ring. More precisely, we prove that the projective bundle formula holds for Heller's relative $K_{0},$ provided $X$ is flat over $S.$ As a corollary, we get a description of the relative group $K_{0}(mathbb{P}_{X}^{r} to mathbb{P}_{S}^{r})$ in terms of generators and relations, provided $X$ is affine and flat over $S.$","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125728825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the number of rational points of smooth projective curves over finite fields in some relative situations in the spirit of a previous paper [HP19] from an euclidean point of vue. We prove some kinds of relative Weil bounds, derived from Schwarz inequality for some "relative parts" of the diagonal and of the graph of the Frobenius on some euclidean sub-spaces of the numerical space of the squared curve endowed with the opposite of the intersection product.
{"title":"Number of points of curves over finite fields in some relative situations from an euclidean point of view","authors":"E. Hallouin, Marc Perret","doi":"10.5802/JTNB.1155","DOIUrl":"https://doi.org/10.5802/JTNB.1155","url":null,"abstract":"We study the number of rational points of smooth projective curves over finite fields in some relative situations in the spirit of a previous paper [HP19] from an euclidean point of vue. We prove some kinds of relative Weil bounds, derived from Schwarz inequality for some \"relative parts\" of the diagonal and of the graph of the Frobenius on some euclidean sub-spaces of the numerical space of the squared curve endowed with the opposite of the intersection product.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123782244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We formulate a geometric Riemann-Hilbert correspondence that applies to the derivation by Jimbo and Sakai of equation $q$-PVI from ``isomonodromy'' conditions. This is a step within work in progress towards the application of $q$-isomonodromy and $q$-isoStokes to $q$-Painleve.
{"title":"The space of monodromy data for the Jimbo–Sakai family of q-difference equations","authors":"Y. Ohyama, J. Ramis, J. Sauloy","doi":"10.5802/AFST.1659","DOIUrl":"https://doi.org/10.5802/AFST.1659","url":null,"abstract":"We formulate a geometric Riemann-Hilbert correspondence that applies to the derivation by Jimbo and Sakai of equation $q$-PVI from ``isomonodromy'' conditions. This is a step within work in progress towards the application of $q$-isomonodromy and $q$-isoStokes to $q$-Painleve.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"12 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131451301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work we study $k$-type uniform Steiner bundles, being $k$ the lowest degree of the splitting. We prove sharp upper and lower bounds for the rank in the case $k=1$ and moreover we give families of examples for every allowed possible rank and explain which relation exists between the families. After dealing with the case $k$ in general, we conjecture that every $k$-type uniform Steiner bundle is obtained through the proposed construction technique.
{"title":"Uniform Steiner bundles","authors":"Simone Marchesi, R. Mir'o-Roig","doi":"10.5802/AIF.3403","DOIUrl":"https://doi.org/10.5802/AIF.3403","url":null,"abstract":"In this work we study $k$-type uniform Steiner bundles, being $k$ the lowest degree of the splitting. We prove sharp upper and lower bounds for the rank in the case $k=1$ and moreover we give families of examples for every allowed possible rank and explain which relation exists between the families. After dealing with the case $k$ in general, we conjecture that every $k$-type uniform Steiner bundle is obtained through the proposed construction technique.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"5 7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123588601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-17DOI: 10.1017/S1743921318005732
G. Cotti
In the first part of this paper, we introduce the notion of "cyclic stratum" of a Frobenius manifold $M$. This is the set of points of the extended manifold $mathbb C^*times M$ at which the unit vector field is a cyclic vector for the isomonodromic system defined by the flatness condition of the extended deformed connection. The study of the geometry of the complement of the cyclic stratum is addressed. We show that at points of the cyclic stratum, the isomonodromic system attached to $M$ can be reduced to a scalar differential equation, called the "master differential equation" of $M$. In the case of Frobenius manifolds coming from Gromov-Witten theory, namely quantum cohomologies of smooth projective varieties, such a construction reproduces the notion of quantum differential equation. In the second part of the paper, we introduce two multilinear transforms, called "Borel-Laplace $(boldsymbol alpha,boldsymbolbeta)$-multitransforms", on spaces of Ribenboim formal power series with exponents and coefficients in an arbitrary finite dimensional $mathbb C$-algebra $A$. When $A$ is specialized to the cohomology of smooth projective varieties, the integral forms of the Borel-Laplace $(boldsymbol alpha,boldsymbolbeta)$-multitransforms are used in order to rephrase the Quantum Lefschetz Theorem. This leads to explicit Mellin-Barnes integral representations of solutions of the quantum differential equations for a wide class of smooth projective varieties, including Fano complete intersections in projective spaces. In the third and final part of the paper, as an application, we show how to use the new analytic tools, introduced in the previous parts, in order to study the quantum differential equations of Hirzebruch surfaces. This finally leads to the proof of Dubrovin Conjecture for all Hirzebruch surfaces.
{"title":"Cyclic stratum of Frobenius manifolds, Borel-Laplace $(boldsymbolalpha,boldsymbolbeta)$-multitransforms, and integral representations of solutions of Quantum Differential Equations","authors":"G. Cotti","doi":"10.1017/S1743921318005732","DOIUrl":"https://doi.org/10.1017/S1743921318005732","url":null,"abstract":"In the first part of this paper, we introduce the notion of \"cyclic stratum\" of a Frobenius manifold $M$. This is the set of points of the extended manifold $mathbb C^*times M$ at which the unit vector field is a cyclic vector for the isomonodromic system defined by the flatness condition of the extended deformed connection. The study of the geometry of the complement of the cyclic stratum is addressed. We show that at points of the cyclic stratum, the isomonodromic system attached to $M$ can be reduced to a scalar differential equation, called the \"master differential equation\" of $M$. In the case of Frobenius manifolds coming from Gromov-Witten theory, namely quantum cohomologies of smooth projective varieties, such a construction reproduces the notion of quantum differential equation. \u0000In the second part of the paper, we introduce two multilinear transforms, called \"Borel-Laplace $(boldsymbol alpha,boldsymbolbeta)$-multitransforms\", on spaces of Ribenboim formal power series with exponents and coefficients in an arbitrary finite dimensional $mathbb C$-algebra $A$. When $A$ is specialized to the cohomology of smooth projective varieties, the integral forms of the Borel-Laplace $(boldsymbol alpha,boldsymbolbeta)$-multitransforms are used in order to rephrase the Quantum Lefschetz Theorem. This leads to explicit Mellin-Barnes integral representations of solutions of the quantum differential equations for a wide class of smooth projective varieties, including Fano complete intersections in projective spaces. \u0000In the third and final part of the paper, as an application, we show how to use the new analytic tools, introduced in the previous parts, in order to study the quantum differential equations of Hirzebruch surfaces. This finally leads to the proof of Dubrovin Conjecture for all Hirzebruch surfaces.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114527447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1974, D. Coray showed that on a smooth cubic surface with a closed point of degree prime to 3 there exists such a point of degree 1, 4 or 10. We first show how a combination of generisation, specialisation, Bertini theorems and large fields avoids considerations of special cases in his argument. For smooth cubic surfaces with a rational point, we show that any zero-cycle of degree at least 10 is rationally equivalent to an effective cycle. We establish analogues of these results for del Pezzo surfaces of degree 2 and of degree 1. For smooth cubic surfaces without a rational point, we relate the question whether there exists a degree 3 point which is not on a line to the question whether rational points are dense on a del Pezzo surface of degree 1. ---- Une surface cubique lisse qui possede un point ferme de degre premier a 3 possede un tel point de degre 1, 4 ou 10 (Coray, 1974). Un melange de generisation, de specialisation, de theoremes de Bertini et d'utilisation des corps fertiles donne de la souplesse a sa methode. Pour les surfaces cubiques avec un point rationnel, on montre que tout zero-cycle de degre au moins 10 est rationnellement equivalent a un zero-cycle effectif. On etablit l'analogue de ces resultats pour les surfaces de del Pezzo de degre 2 et de degre 1. On discute l'existence de points fermes de degre 3 non alignes sur une surface cubique sans point rationnel. On la relie a la question de la densite des points rationnels sur une surface de del Pezzo de degre 1.
{"title":"Zéro-cycles sur les surfaces de del Pezzo (Variations sur un thème de Daniel Coray)","authors":"Jean-Louis Colliot-Th'elene","doi":"10.4171/LEM/66-3/4-8","DOIUrl":"https://doi.org/10.4171/LEM/66-3/4-8","url":null,"abstract":"In 1974, D. Coray showed that on a smooth cubic surface with a closed point of degree prime to 3 there exists such a point of degree 1, 4 or 10. We first show how a combination of generisation, specialisation, Bertini theorems and large fields avoids considerations of special cases in his argument. For smooth cubic surfaces with a rational point, we show that any zero-cycle of degree at least 10 is rationally equivalent to an effective cycle. We establish analogues of these results for del Pezzo surfaces of degree 2 and of degree 1. For smooth cubic surfaces without a rational point, we relate the question whether there exists a degree 3 point which is not on a line to the question whether rational points are dense on a del Pezzo surface of degree 1. \u0000---- \u0000Une surface cubique lisse qui possede un point ferme de degre premier a 3 possede un tel point de degre 1, 4 ou 10 (Coray, 1974). Un melange de generisation, de specialisation, de theoremes de Bertini et d'utilisation des corps fertiles donne de la souplesse a sa methode. Pour les surfaces cubiques avec un point rationnel, on montre que tout zero-cycle de degre au moins 10 est rationnellement equivalent a un zero-cycle effectif. On etablit l'analogue de ces resultats pour les surfaces de del Pezzo de degre 2 et de degre 1. On discute l'existence de points fermes de degre 3 non alignes sur une surface cubique sans point rationnel. On la relie a la question de la densite des points rationnels sur une surface de del Pezzo de degre 1.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134281571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-06DOI: 10.4310/CNTP.2021.v15.n3.a1
G. Denham, Delphine Pol, M. Schulze, U. Walther
Generalizing the star graphs of Muller-Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a non-trivial torus action. For such graphs, we show that the Euler characteristic of the corresponding projective graph hypersurface complement is zero. In contrast, we also show that the Euler characteristic in question can take any integer value for a suitable graph. This disproves a conjecture of Aluffi in a strong sense.
{"title":"Graph hypersurfaces with torus action and a conjecture of Aluffi","authors":"G. Denham, Delphine Pol, M. Schulze, U. Walther","doi":"10.4310/CNTP.2021.v15.n3.a1","DOIUrl":"https://doi.org/10.4310/CNTP.2021.v15.n3.a1","url":null,"abstract":"Generalizing the star graphs of Muller-Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a non-trivial torus action. For such graphs, we show that the Euler characteristic of the corresponding projective graph hypersurface complement is zero. In contrast, we also show that the Euler characteristic in question can take any integer value for a suitable graph. This disproves a conjecture of Aluffi in a strong sense.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127428686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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