We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E E , where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E → P 1 Eto mathbb {P}^1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.
{"title":"Enumerating pencils with moving ramification on curves","authors":"Carl Lian","doi":"10.1090/jag/776","DOIUrl":"https://doi.org/10.1090/jag/776","url":null,"abstract":"We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve \u0000\u0000 \u0000 E\u0000 E\u0000 \u0000\u0000, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps \u0000\u0000 \u0000 \u0000 E\u0000 →\u0000 \u0000 \u0000 P\u0000 \u0000 1\u0000 \u0000 \u0000 Eto mathbb {P}^1\u0000 \u0000\u0000 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43209329","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 construct a generalisation of Mantovan’s almost product structure to Shimura varieties of Hodge type with hyperspecial level structure at p p and deduce that the perfection of the Newton strata are proétale locally isomorphic to the perfection of the product of a central leaf and a Rapoport-Zink space. The almost product formula can be extended to obtain an analogue of Caraiani and Scholze’s generalisation of the almost product structure for Shimura varieties of Hodge type.
{"title":"The product structure of Newton strata in the good reduction of Shimura varieties of Hodge type","authors":"Paul Hamacher","doi":"10.1090/JAG/732","DOIUrl":"https://doi.org/10.1090/JAG/732","url":null,"abstract":"We construct a generalisation of Mantovan’s almost product structure to Shimura varieties of Hodge type with hyperspecial level structure at \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000 and deduce that the perfection of the Newton strata are proétale locally isomorphic to the perfection of the product of a central leaf and a Rapoport-Zink space. The almost product formula can be extended to obtain an analogue of Caraiani and Scholze’s generalisation of the almost product structure for Shimura varieties of Hodge type.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/732","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47307786","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}
Asher Auel, C. Böhning, H. G. Bothmer, Alena Pirutka
We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P3 where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen–Macaulay sheaves, as well as the geometry of special arrangements of rational curves in P2. We also prove the existence of universally CH0-trivial resolutions for the general class of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.
{"title":"Conic bundle fourfolds with nontrivial unramified Brauer group","authors":"Asher Auel, C. Böhning, H. G. Bothmer, Alena Pirutka","doi":"10.1090/jag/743","DOIUrl":"https://doi.org/10.1090/jag/743","url":null,"abstract":"We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P3 where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen–Macaulay sheaves, as well as the geometry of special arrangements of rational curves in P2. We also prove the existence of universally CH0-trivial resolutions for the general class of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/743","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46647036","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 $X$ be a smooth projective variety. The Iitaka dimension of a divisor $D$ is an important invariant, but it does not only depend on the numerical class of $D$. However, there are several definitions of ``numerical Iitaka dimension'', depending only on the numerical class. In this note, we show that there exists a pseuodoeffective $mathbb R$-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective $mathbb R$-divisor $D_+$ for which $h^0(X,lfloor m D_+ rfloor+A)$ is bounded above and below by multiples of $m^{3/2}$ for any sufficiently ample $A$.
设$X$是一个光滑的投影变量。除数$D$的Iitaka维数是一个重要的不变量，但它不仅取决于$D$的数值类。然而，有几种“数值Iitaka维”的定义，仅取决于数值类。在本文中，我们证明存在一个伪有效的$mathbb R$除数，其中这些不变量取不同的值。关键是构造一个伪有效的$mathbb R$-除数$D_+$的例子，其中$h^0(X，lfloor m D_+ rfloor+ a)$上下以$m^{3/2}$的倍数为界，对于任何足够充裕的$ a $。
{"title":"Notions of numerical Iitaka dimension do not coincide","authors":"John Lesieutre","doi":"10.1090/JAG/763","DOIUrl":"https://doi.org/10.1090/JAG/763","url":null,"abstract":"Let $X$ be a smooth projective variety. The Iitaka dimension of a divisor $D$ is an important invariant, but it does not only depend on the numerical class of $D$. However, there are several definitions of ``numerical Iitaka dimension'', depending only on the numerical class. In this note, we show that there exists a pseuodoeffective $mathbb R$-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective $mathbb R$-divisor $D_+$ for which $h^0(X,lfloor m D_+ rfloor+A)$ is bounded above and below by multiples of $m^{3/2}$ for any sufficiently ample $A$.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45264054","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 paper, we expand the foundations of derived complex analytic geometry introduced by Jacob Lurie in 2011. We start by studying the analytification functor and its properties. In particular, we prove that for a derived complex scheme locally almost of finite presentation X X , the canonical map X a n → X X^{mathrm {an}} to X is flat in the derived sense. Next, we provide a comparison result relating derived complex analytic spaces to geometric stacks. Using these results and building on the previous work of the author and Tony Yue Yu, we prove a derived version of the GAGA theorems. As an application, we prove that the infinitesimal deformation theory of a derived complex analytic moduli problem is governed by a differential graded Lie algebra.
{"title":"GAGA theorems in derived complex geometry","authors":"Mauro Porta","doi":"10.1090/JAG/716","DOIUrl":"https://doi.org/10.1090/JAG/716","url":null,"abstract":"In this paper, we expand the foundations of derived complex analytic geometry introduced by Jacob Lurie in 2011. We start by studying the analytification functor and its properties. In particular, we prove that for a derived complex scheme locally almost of finite presentation \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000, the canonical map \u0000\u0000 \u0000 \u0000 \u0000 X\u0000 \u0000 \u0000 a\u0000 n\u0000 \u0000 \u0000 \u0000 →\u0000 X\u0000 \u0000 X^{mathrm {an}} to X\u0000 \u0000\u0000 is flat in the derived sense. Next, we provide a comparison result relating derived complex analytic spaces to geometric stacks. Using these results and building on the previous work of the author and Tony Yue Yu, we prove a derived version of the GAGA theorems. As an application, we prove that the infinitesimal deformation theory of a derived complex analytic moduli problem is governed by a differential graded Lie algebra.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/716","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47412219","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 give a closed formula for the Néron–Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number ω ^ 2 hat {omega }^2 of the dualizing sheaf of a curve in terms of Zhang’s invariant φ varphi . As an application, we obtain an effective Bogomolov-type result for the tautological cycles. We deduce these results from a more general combinatorial computation of arithmetic intersection numbers of adelic line bundles on higher self-products of curves, which are linear combinations of pullbacks of line bundles on the curve and the diagonal bundle.
{"title":"On arithmetic intersection numbers on self-products of curves","authors":"R. Wilms","doi":"10.1090/jag/777","DOIUrl":"https://doi.org/10.1090/jag/777","url":null,"abstract":"We give a closed formula for the Néron–Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number \u0000\u0000 \u0000 \u0000 \u0000 \u0000 ω\u0000 ^\u0000 \u0000 \u0000 2\u0000 \u0000 hat {omega }^2\u0000 \u0000\u0000 of the dualizing sheaf of a curve in terms of Zhang’s invariant \u0000\u0000 \u0000 φ\u0000 varphi\u0000 \u0000\u0000. As an application, we obtain an effective Bogomolov-type result for the tautological cycles. We deduce these results from a more general combinatorial computation of arithmetic intersection numbers of adelic line bundles on higher self-products of curves, which are linear combinations of pullbacks of line bundles on the curve and the diagonal bundle.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45289400","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}
Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on algebraic hyperbolicity of very general surfaces in toric threefolds.
{"title":"Algebraic hyperbolicity for surfaces in toric threefolds","authors":"Christian Haase, N. Ilten","doi":"10.1090/JAG/770","DOIUrl":"https://doi.org/10.1090/JAG/770","url":null,"abstract":"Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on algebraic hyperbolicity of very general surfaces in toric threefolds.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42274765","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 use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degree 36 and of an irregular surface with canonical map of degree 27. As a by-product, we get equations (over a finite field) for the Z / 3 mathbb {Z}/3 -invariant fibres of the Albanese fibration of the Cartwright-Steger surface and show that they are smooth.
{"title":"Surfaces with canonical map of maximum degree","authors":"Carlos Rito","doi":"10.1090/JAG/761","DOIUrl":"https://doi.org/10.1090/JAG/761","url":null,"abstract":"We use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degree 36 and of an irregular surface with canonical map of degree 27. As a by-product, we get equations (over a finite field) for the \u0000\u0000 \u0000 \u0000 \u0000 Z\u0000 \u0000 \u0000 /\u0000 \u0000 3\u0000 \u0000 mathbb {Z}/3\u0000 \u0000\u0000-invariant fibres of the Albanese fibration of the Cartwright-Steger surface and show that they are smooth.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44853823","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}
<p>We construct a compactification of the moduli space of Drinfeld modules of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and level <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a moduli space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-reciprocal maps. This is closely related to the Satake compactification but not exactly the same. The construction involves some technical assumptions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that are satisfied for a cofinal set of ideals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the special case where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals double-struck upper F Subscript q Baseline left-bracket t right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A=mathbb {F}_q[t]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N equals left-parenthesis t Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">N=(t^n)</mml:annotation> </mml:semanti
{"title":"Compactification of Drinfeld moduli spaces as moduli spaces of 𝐴-reciprocal maps and consequences for Drinfeld modular forms","authors":"R. Pink","doi":"10.1090/jag/772","DOIUrl":"https://doi.org/10.1090/jag/772","url":null,"abstract":"<p>We construct a compactification of the moduli space of Drinfeld modules of rank <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r\">\u0000 <mml:semantics>\u0000 <mml:mi>r</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">r</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and level <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\">\u0000 <mml:semantics>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">N</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> as a moduli space of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-reciprocal maps. This is closely related to the Satake compactification but not exactly the same. The construction involves some technical assumptions on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\">\u0000 <mml:semantics>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">N</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> that are satisfied for a cofinal set of ideals <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\">\u0000 <mml:semantics>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">N</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In the special case where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A equals double-struck upper F Subscript q Baseline left-bracket t right-bracket\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>q</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">A=mathbb {F}_q[t]</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N equals left-parenthesis t Superscript n Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">N=(t^n)</mml:annotation>\u0000 </mml:semanti","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45193739","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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