H. Markwig, Thomas Markwig, Kristin M. Shaw, E. Shustin
{"title":"Tropical floor plans and enumeration of complex and real multi-nodal surfaces","authors":"H. Markwig, Thomas Markwig, Kristin M. Shaw, E. Shustin","doi":"10.1090/jag/774","DOIUrl":null,"url":null,"abstract":"<p>The family of complex projective surfaces 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 </mml:semantics>\n</mml:math>\n</inline-formula> of degree <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> having precisely <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<!-- 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<!-- 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<!-- 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<!-- 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<!-- 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<!-- 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 {P}^3}(d)=(4(d-1)^3)^\\delta /\\delta !+O(d^{3\\delta -3})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. 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<!-- 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<!-- 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<!-- 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<!-- 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<!-- 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 P3\mathbb {P}^3 of degree dd having precisely δ\delta nodes as their only singularities has codimension δ\delta in the linear system |OP3(d)||{\mathcal O}_{\mathbb {P}^3}(d)| for sufficiently large dd and is of degree Nδ,CP3(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\delta -3}). In particular, Nδ,CP3(d)N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d) is polynomial in dd.
By means of tropical geometry, we explicitly describe (4d3)δ/δ!+O(d3δ−1)(4d^3)^\delta /\delta !+O(d^{3\delta -1}) surfaces passing through a suitable generic configuration of n=(d+33)−δ−1n=\binom {d+3}{3}-\delta -1 points in P3\mathbb {P}^3
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.