Tropical floor plans and enumeration of complex and real multi-nodal surfaces

IF 0.9 1区数学 Q2 MATHEMATICS

Journal of Algebraic Geometry Pub Date : 2019-10-18 DOI:10.1090/jag/774

H. Markwig, Thomas Markwig, Kristin M. Shaw, E. Shustin

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<mml:semantics>\n <mml:mi>δ</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\delta</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> nodes as their only singularities has codimension <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta\">\n <mml:semantics>\n <mml:mi>δ</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\delta</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in the linear system <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue script upper O Subscript double-struck upper P cubed Baseline left-parenthesis d right-parenthesis EndAbsoluteValue\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">P</mml:mi>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:msup>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>d</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">|{\\mathcal O}_{\\mathbb {P}^3}(d)|</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for sufficiently large <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\n <mml:semantics>\n <mml:mi>d</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and is of degree <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N Subscript delta comma double-struck upper C Superscript double-struck upper P cubed Baseline left-parenthesis d right-parenthesis equals left-parenthesis 4 left-parenthesis d minus 1 right-parenthesis cubed right-parenthesis Superscript delta Baseline slash delta factorial plus upper O left-parenthesis d Superscript 3 delta minus 3 Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msubsup>\n <mml:mi>N</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>δ</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">P</mml:mi>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:msup>\n </mml:mrow>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>d</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>4</mml:mn>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>d</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mn>3</mml:mn>\n </mml:msup>\n <mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mi>δ</mml:mi>\n </mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mi>δ</mml:mi>\n <mml:mo>!</mml:mo>\n <mml:mo>+</mml:mo>\n <mml:mi>O</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>d</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>3</mml:mn>\n <mml:mi>δ</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mn>3</mml:mn>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">N_{\\delta ,\\mathbb {C}}^{\\mathbb 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In particular, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N Subscript delta comma double-struck upper C Superscript double-struck upper P cubed Baseline left-parenthesis d right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msubsup>\n <mml:mi>N</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>δ</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">P</mml:mi>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:msup>\n </mml:mrow>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>d</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">N_{\\delta ,\\mathbb {C}}^{\\mathbb {P}^3}(d)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is polynomial in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\n <mml:semantics>\n <mml:mi>d</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>\n\n<p>By means of tropical geometry, we explicitly describe <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis 4 d cubed right-parenthesis Superscript delta Baseline slash delta factorial plus upper O left-parenthesis d Superscript 3 delta minus 1 Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>4</mml:mn>\n <mml:msup>\n <mml:mi>d</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msup>\n <mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mi>δ</mml:mi>\n </mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mi>δ</mml:mi>\n <mml:mo>!</mml:mo>\n <mml:mo>+</mml:mo>\n <mml:mi>O</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>d</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>3</mml:mn>\n <mml:mi>δ</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(4d^3)^\\delta /\\delta !+O(d^{3\\delta -1})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> surfaces passing through a suitable generic configuration of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n equals StartBinomialOrMatrix d plus 3 Choose 3 EndBinomialOrMatrix minus delta minus 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>n</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mrow>\n <mml:mstyle scriptlevel=\"0\">\n <mml:mrow class=\"MJX-TeXAtom-OPEN\">\n <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo>\n </mml:mrow>\n </mml:mstyle>\n <mml:mfrac linethickness=\"0\">\n <mml:mrow>\n <mml:mi>d</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>3</mml:mn>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:mfrac>\n <mml:mstyle scriptlevel=\"0\">\n <mml:mrow class=\"MJX-TeXAtom-CLOSE\">\n <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo>\n </mml:mrow>\n </mml:mstyle>\n </mml:mrow>\n <mml:mo>−</mml:mo>\n <mml:mi>δ</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">n=\\binom {d+3}{3}-\\delta -1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> points in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P cubed\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">P</mml:mi>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {P}^3</mml:annotation>\n </mm","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2019-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jag/774","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 1

Abstract

The family of complex projective surfaces in P 3 \mathbb {P}^3 of degree d d having precisely δ \delta nodes as their only singularities has codimension δ \delta in the linear system | O P 3 ( d ) | |{\mathcal O}_{\mathbb {P}^3}(d)| for sufficiently large d d and is of degree N δ , C P 3 ( d ) = ( 4 ( d − 1 ) 3 ) δ / δ ! + O ( d 3 δ − 3 ) N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d)=(4(d-1)^3)^\delta /\delta !+O(d^{3\delta -3}) . In particular, N δ , C P 3 ( d ) N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d) is polynomial in d d .

By means of tropical geometry, we explicitly describe ( 4 d 3 ) δ / δ ! + O ( d 3 δ − 1 ) (4d^3)^\delta /\delta !+O(d^{3\delta -1}) surfaces passing through a suitable generic configuration of n = ( d + 3 3 ) − δ − 1 n=\binom {d+3}{3}-\delta -1 points in P 3 \mathbb {P}^3

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热带楼层平面图和复杂和真实多节点表面的列举

在线性系统|O P中，具有精确的δδ节点作为其唯一奇点的P3\mathbb｛P｝^3中的复射影曲面族具有余维δ3（d）||{\mathcal O}_{\mathbb{P}^3}（d）|，C P 3（d）=（4（d−1）3）δ/δ！+O（d3δ−3）N_｛\delta，\mathbb｛C｝｝^｛\mathbb{P｝^3｝（d）=（4（d-1）^3）^\ delta/\delta+O（d^｛3\Δ-3｝）。特别地，Nδ，C P3（d）N_｛\delta，\mathbb｛C｝｝^｛\mathbb{P｝^3｝（d）是d d中的多项式，我们明确地描述了（4d3）δ/δ！+O（d3δ−1）（4d^3）^\delta/\delta+O（d^｛3\delta-1｝）表面穿过n＝（d+3 3）−δ−1 n＝\binom｛d+3｝的合适的一般构型{3}-\P 3\mathbb｛P｝^3中的delta-1点

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来源期刊

Journal of Algebraic Geometry 数学-数学

CiteScore

2.70

自引率

5.60%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology. This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.

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