Consider a log Calabi-Yau pair (X,D)(X,D) consisting of a smooth del Pezzo surface XX of degree ≥3geq 3 and a smooth anticanonical divisor DD. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of XX intersecting DD in a single point with maximal tangency and the consistent wall structure appearing in the dual intersection complex of (X,D)(X,D) from the Gross-Siebert reconstruction algorithm. More precisely, the logarithm of the product of functions attached to unbounded walls in the consistent wall structure gives a generati
{"title":"Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs","authors":"Tim Graefnitz","doi":"10.1090/jag/794","DOIUrl":"https://doi.org/10.1090/jag/794","url":null,"abstract":"<p>Consider a log Calabi-Yau pair <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper X comma upper D right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(X,D)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> consisting of a smooth del Pezzo surface <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of degree <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"greater-than-or-equal-to 3\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">geq 3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and a smooth anticanonical divisor <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\u0000 <mml:semantics>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> intersecting <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\u0000 <mml:semantics>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in a single point with maximal tangency and the consistent wall structure appearing in the dual intersection complex of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper X comma upper D right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(X,D)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> from the Gross-Siebert reconstruction algorithm. More precisely, the logarithm of the product of functions attached to unbounded walls in the consistent wall structure gives a generati","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48480911","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 prove that the ramified Prym map Pg,rmathcal P_{g, r} which sends a covering π:D⟶Cpi :Dlongrightarrow C ramified in rr points to the Prym variety P(π)≔Ker(Nmπ)P(pi )≔Ker(Nm_{pi }) is an embedding for all r≥6rge 6 and for all
我们证明了分支Prym映射P g, r mathcal p＿{G, r} 它发出一个覆盖π: D pi : dlongrightarrow C在r中的分支r指向P(π)，其中K e r (N m π) P(pi )对象是Ker(Nm＿{pi })是对所有r≥6r的嵌入ge 对于所有g(C)>0 g(C)>0。此外，通过研究超椭圆曲线覆盖轨迹的限制，我们证明了P g, 2 mathcal p＿{G, 2} P g, 4 mathcal p＿{G, 4} 有正维纤维。
{"title":"Global Prym-Torelli for double coverings ramified in at least six points","authors":"J. Naranjo, –. Ortega","doi":"10.1090/jag/779","DOIUrl":"https://doi.org/10.1090/jag/779","url":null,"abstract":"<p>We prove that the ramified Prym map <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper P Subscript g comma r\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>g</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>r</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal P_{g, r}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> which sends a covering <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi colon upper D long right-arrow upper C\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mo>:</mml:mo>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">⟶<!-- ⟶ --></mml:mo>\u0000 <mml:mi>C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">pi :Dlongrightarrow C</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> ramified in <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> points to the Prym variety <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P left-parenthesis pi right-parenthesis colon-equal upper K e r left-parenthesis upper N m Subscript pi Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>P</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>≔</mml:mo>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>e</mml:mi>\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">P(pi )≔Ker(Nm_{pi })</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is an embedding for all <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r greater-than-or-equal-to 6\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>6</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">rge 6</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and for all <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g left-parenthesis upper C ri","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45359188","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 study the relative anti-canonical divisor − K X / Y -K_{X/Y} of an algebraic fiber space ϕ : X → Y phi colon Xto Y , and we reveal relations among positivity conditions of − K X / Y -K_{X/Y} , certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana–Cao–Matsumura’s equality on Hacon–McKernan’s question, whose original proof depends on analytics methods. The third result proves that algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite étale cover of Y Y . Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.
{"title":"On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces","authors":"Sho Ejiri, M. Iwai, Shin-ichi Matsumura","doi":"10.1090/jag/814","DOIUrl":"https://doi.org/10.1090/jag/814","url":null,"abstract":"In this paper, we study the relative anti-canonical divisor \u0000\u0000 \u0000 \u0000 −\u0000 \u0000 K\u0000 \u0000 X\u0000 \u0000 /\u0000 \u0000 Y\u0000 \u0000 \u0000 \u0000 -K_{X/Y}\u0000 \u0000\u0000 of an algebraic fiber space \u0000\u0000 \u0000 \u0000 ϕ\u0000 :\u0000 X\u0000 →\u0000 Y\u0000 \u0000 phi colon Xto Y\u0000 \u0000\u0000, and we reveal relations among positivity conditions of \u0000\u0000 \u0000 \u0000 −\u0000 \u0000 K\u0000 \u0000 X\u0000 \u0000 /\u0000 \u0000 Y\u0000 \u0000 \u0000 \u0000 -K_{X/Y}\u0000 \u0000\u0000, certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana–Cao–Matsumura’s equality on Hacon–McKernan’s question, whose original proof depends on analytics methods. The third result proves that algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite étale cover of \u0000\u0000 \u0000 Y\u0000 Y\u0000 \u0000\u0000. Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45585372","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}
Suppose that XX is a projective manifold whose tangent bundle TXT_X contains a locally free strictly nef subsheaf. We prove that XX is isomorphic to either a projective space or a projective bundle over a hyperbolic manifold of general type. Moreover, if the fundamental group π1(X)pi _1(X) is virtually solvable, then XX is isomorphic to a projective space.
设X X是一个射影流形，其切束T X T_X包含一个局部自由的严格nef子轴。证明X X与一般型双曲流形上的射影空间或射影束同构。此外，如果基本群π 1(X) pi _1(X)是虚可解的，则X X是射影空间同构的。
{"title":"Projective manifolds whose tangent bundle contains a strictly nef subsheaf","authors":"Jie Liu, Wenhao Ou, Xiaokui Yang","doi":"10.1090/jag/807","DOIUrl":"https://doi.org/10.1090/jag/807","url":null,"abstract":"<p>Suppose that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a projective manifold whose tangent bundle <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T Subscript upper X\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:mi>X</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">T_X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> contains a locally free strictly nef subsheaf. We prove that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is isomorphic to either a projective space or a projective bundle over a hyperbolic manifold of general type. Moreover, if the fundamental group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi 1 left-parenthesis upper X right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">pi _1(X)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is virtually solvable, then <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is isomorphic to a projective space.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43641826","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 Brill-Noether locus M18,163M^3_{18,16} is an irreducible component of the Gieseker-Petri locus in genus 1818 having codimension 22 in the moduli space of curves. This result disproves a conjecture predicting that the Gieseker-Petri locus is always divisorial.
{"title":"A codimension 2 component of the Gieseker-Petri locus","authors":"Margherita Lelli–Chiesa","doi":"10.1090/jag/780","DOIUrl":"https://doi.org/10.1090/jag/780","url":null,"abstract":"<p>We show that the Brill-Noether locus <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M Subscript 18 comma 16 Superscript 3\">\u0000 <mml:semantics>\u0000 <mml:msubsup>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>18</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>16</mml:mn>\u0000 </mml:mrow>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msubsup>\u0000 <mml:annotation encoding=\"application/x-tex\">M^3_{18,16}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is an irreducible component of the Gieseker-Petri locus in genus <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"18\">\u0000 <mml:semantics>\u0000 <mml:mn>18</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">18</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> having codimension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in the moduli space of curves. This result disproves a conjecture predicting that the Gieseker-Petri locus is always divisorial.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47940720","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 XX be a smooth, projective, rationally connected variety, defined over a number field kk, and let Z⊂XZsubset X be a closed subset of codimension at least two. In this paper, for certain choices of XX, we prove that the set of ZZ-integral points is potentially Zariski dense, in the sense that there is a finite extension KK of kk such that the set of points P∈X(K)Pin X(K) that are
设X X是定义在数域k k上的光滑的、射影的、合理连通的变种，并设Z∧X Z子集X是余维至少为2的闭子集。在本文中，对于X X的某些选择，我们证明了Z Z积分点的集合是潜在的Zariski稠密的，即K K的有限扩展K K使得点P∈X(K) P In X(K)是Z Z积分的集合P∈X(K) P In X(X)是Zariski稠密的。这对哈塞特和茨钦克尔2001年提出的一个问题给出了肯定的答案。
{"title":"Codimension two integral points on some rationally connected threefolds are potentially dense","authors":"David McKinnon, Mike Roth","doi":"10.1090/jag/782","DOIUrl":"https://doi.org/10.1090/jag/782","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a smooth, projective, rationally connected variety, defined over a number field <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z subset-of upper X\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>Z</mml:mi>\u0000 <mml:mo>⊂<!-- ⊂ --></mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Zsubset X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a closed subset of codimension at least two. In this paper, for certain choices of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, we prove that the set of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z\">\u0000 <mml:semantics>\u0000 <mml:mi>Z</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Z</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-integral points is potentially Zariski dense, in the sense that there is a finite extension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> such that the set of points <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P element-of upper X left-parenthesis upper K right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>P</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Pin X(K)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> that are <inline-formula content-type=\"math/mathml\">\u0000<mml:math x","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48063382","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 prove a conjecture of Markman about the shape of the monodromy group of irreducible holomorphic symplectic manifolds of OG10 type. As a corollary, we also compute the locally trivial monodromy group of the underlying singular symplectic variety.
{"title":"On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces","authors":"C. Onorati","doi":"10.1090/jag/802","DOIUrl":"https://doi.org/10.1090/jag/802","url":null,"abstract":"In this paper we prove a conjecture of Markman about the shape of the monodromy group of irreducible holomorphic symplectic manifolds of OG10 type. As a corollary, we also compute the locally trivial monodromy group of the underlying singular symplectic variety.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49367741","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 prove Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse Q¯ℓoverline {mathbb {Q}}_{ell }-sheaves. This was originally solved by Kerz and Saito in characteristic ≠2neq 2.
{"title":"Bloch’s formula for 0-cycles with modulus and higher-dimensional class field theory","authors":"F. Binda, A. Krishna, S. Saito","doi":"10.1090/jag/792","DOIUrl":"https://doi.org/10.1090/jag/792","url":null,"abstract":"<p>We prove Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q overbar Subscript script l\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo>\u0000 </mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>ℓ<!-- ℓ --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">overline {mathbb {Q}}_{ell }</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-sheaves. This was originally solved by Kerz and Saito in characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"not-equals 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo>≠<!-- ≠ --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">neq 2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44914700","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}
H. Markwig, Thomas Markwig, Kristin M. Shaw, E. Shustin
The family of complex projective surfaces in P3mathbb {P}^3 of degree dd having precisely δdelta nodes as their only singularities has codimension δdelta in the linear system |OP3(d)||{mathcal O}_{mathbb {P}^3}(d)| for sufficiently large dd
Let XX be a smooth projective variety over an algebraically closed field, and f:X→Xfcolon Xto X a surjective self-morphism of XX. The ii-th cohomological dynamical degree χi(f)chi _i(f) is defined as the spectral radius of the pullback f∗f^{*} on the étale cohomology group H
设X X是代数闭域上的光滑射影变，且f: X→X f colon X to X是X X的满射自态射。i i -上同调动力学度χ i(f) chi ＿i(f)定义为在上同调群H上的回拉f {* f^*}的谱半径。Q (l) H^i＿ {acute{mathrm e{}}mathrm t{(X，}}mathbf Q＿ {}ell)和k k -数值动力度λ k(f) lambda ＿k(f)作为回拉f {* f^*}在向量空间N k(X) R mathsf N{^k(X)＿ }{mathbf R{上的谱半径余维k k在X X模数值等价上的代数循环。作为Weil黎曼假设的推广，Truong推测χ }}2k(f) = λ k(f) chi ＿2k(f) = {}lambda ＿k(f)对于所有0≤k≤dim (X) 0 le k ledim X。我们在阿贝尔变的情况下证明了这个猜想。在证明过程中，我们还得到了关于素数特征的阿贝尔变体的自映射的特征值的一个新的奇偶性结果，这是一个独立的研究方向。
{"title":"Eigenvalues and dynamical degrees of self-maps on abelian varieties","authors":"Fei Hu","doi":"10.1090/jag/806","DOIUrl":"https://doi.org/10.1090/jag/806","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a smooth projective variety over an algebraically closed field, and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon upper X right-arrow upper X\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo>:<!-- : --></mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">fcolon Xto X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> a surjective self-morphism of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"i\">\u0000 <mml:semantics>\u0000 <mml:mi>i</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">i</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-th cohomological dynamical degree <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"chi Subscript i Baseline left-parenthesis f right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>χ<!-- χ --></mml:mi>\u0000 <mml:mi>i</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">chi _i(f)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is defined as the spectral radius of the pullback <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">f^{*}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> on the étale cohomology group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Subscript ModifyingAbove normal e With acute normal t Superscript i Baseline left-parenthesis upper X comma bold upper Q Subscript script l Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:move","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46819321","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：