Deformation and Unobstructedness of Determinantal Schemes

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2020-07-23 DOI:10.1090/memo/1418

Jan O. Kleppe, R. Mir'o-Roig

{"title":"Deformation and Unobstructedness of Determinantal Schemes","authors":"Jan O. Kleppe, R. Mir'o-Roig","doi":"10.1090/memo/1418","DOIUrl":null,"url":null,"abstract":"<p>A closed subscheme <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X subset-of double-struck upper P Superscript n\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>X</mml:mi>\n <mml:mo>⊂</mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">P</mml:mi>\n </mml:mrow>\n <mml:mi>n</mml:mi>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">X\\subset \\mathbb {P}^n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is said to be <italic>determinantal</italic> if its homogeneous saturated ideal can be generated by the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"s times s\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>s</mml:mi>\n <mml:mo>×</mml:mo>\n <mml:mi>s</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">s\\times s</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> minors of a homogeneous <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p times q\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>p</mml:mi>\n <mml:mo>×</mml:mo>\n <mml:mi>q</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">p\\times q</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> matrix satisfying <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis p minus s plus 1 right-parenthesis left-parenthesis q minus s plus 1 right-parenthesis equals n minus dimension upper X\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>p</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mi>s</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>q</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mi>s</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mi>n</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mi>dim</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mi>X</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(p-s+1)(q-s+1)=n - \\dim X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and it is said to be <italic>standard determinantal</italic> if, in addition, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"s equals min left-parenthesis p comma q right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>s</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mo movablelimits=\"true\" form=\"prefix\">min</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>p</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>q</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">s=\\min (p,q)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Given integers <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"a 1 less-than-or-equal-to a 2 less-than-or-equal-to midline-horizontal-ellipsis less-than-or-equal-to a Subscript t plus c minus 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>a</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo>≤</mml:mo>\n <mml:msub>\n <mml:mi>a</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo>≤</mml:mo>\n <mml:mo>⋯</mml:mo>\n <mml:mo>≤</mml:mo>\n <mml:msub>\n <mml:mi>a</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>t</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mi>c</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">a_1\\le a_2\\le \\cdots \\le a_{t+c-1}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"b 1 less-than-or-equal-to b 2 less-than-or-equal-to midline-horizontal-ellipsis less-than-or-equal-to b Subscript t\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>b</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo>≤</mml:mo>\n <mml:msub>\n <mml:mi>b</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo>≤</mml:mo>\n <mml:mo>⋯</mml:mo>\n <mml:mo>≤</mml:mo>\n <mml:msub>\n <mml:mi>b</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">b_1\\le b_2 \\le \\cdots \\le b_t</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> we consider <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t times left-parenthesis t plus c minus 1 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>t</mml:mi>\n <mml:mo>×</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mi>c</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">t\\times (t+c-1)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> matrices <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper A equals left-parenthesis f Subscript i j Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A</mml:mi>\n </mml:mrow>\n <mml:mo>=</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>f</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>i</mml:mi>\n <mml:mi>j</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {A}=(f_{ij})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with entries homogeneous forms of degree <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"a Subscript j Baseline minus b Subscript i\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>a</mml:mi>\n <mml:mi>j</mml:mi>\n </mml:msub>\n <mml:mo>−</mml:mo>\n <mml:msub>\n <mml:mi>b</mml:mi>\n <mml:mi>i</mml:mi>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">a_j-b_i</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and we denote by <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper W left-parenthesis b underbar semicolon a underbar semicolon r right-parenthesis With bar\">\n <mml:semantics>\n <mml:mover>\n <mml:mrow>\n <mml:mi>W</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:munder>\n <mml:mi>b</mml:mi>\n <mml:mo>_</mml:mo>\n </mml:munder>\n <mml:mo>;</mml:mo>\n <mml:munder>\n <mml:mi>a</mml:mi>\n <mml:mo>_</mml:mo>\n </mml:munder>\n <mml:mo>;</mml:mo>\n <mml:mi>r</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:mo accent=\"false\">¯</mml:mo>\n </mml:mover>\n <mml:annotation encoding=\"application/x-tex\">\\overline {W(\\underline {b};\\underline {a};r)}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> the closure of the locus <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper W left-parenthesis b underbar semicolon a underbar semicolon r right-parenthesis subset-of upper H i l b Superscript p left-parenthesis t right-parenthesis Baseline left-parenthesis double-struck upper P Superscript n Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>W</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:munder>\n <mml:mi>b</mml:mi>\n <mml:mo>_</mml:mo>\n </mml:munder>\n <mml:mo>;</mml:mo>\n <mml:munder>\n <mml:mi>a</mml:mi>\n <mml:mo>_</mml:mo>\n </mml:munder>\n <mml:mo>;</mml:mo>\n <mml:mi>r</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>⊂</mml:mo>\n <mml:mi>H</mml:mi>\n <mml:mi>i</mml:mi>\n <mml:mi>l</mml:mi>\n <mml:msup>\n <mml:mi>b</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>p</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">P</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>n</mml:mi>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">W(\\underline {b};\\underline {a};r)\\subset Hilb^{p(t)}(\\mathbb {P}^{n})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of determinantal schemes defined by the vanishing of the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/memo/1418","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}

引用次数: 2

Abstract

A closed subscheme X ⊂ P n X\subset \mathbb {P}^n is said to be determinantal if its homogeneous saturated ideal can be generated by the s × s s\times s minors of a homogeneous p × q p\times q matrix satisfying ( p − s + 1 ) ( q − s + 1 ) = n − dim ⁡ X (p-s+1)(q-s+1)=n - \dim X and it is said to be standard determinantal if, in addition, s = min ( p , q ) s=\min (p,q) . Given integers a 1 ≤ a 2 ≤ ⋯ ≤ a t + c − 1 a_1\le a_2\le \cdots \le a_{t+c-1} and b 1 ≤ b 2 ≤ ⋯ ≤ b t b_1\le b_2 \le \cdots \le b_t we consider t × ( t + c − 1 ) t\times (t+c-1) matrices A = ( f i j ) \mathcal {A}=(f_{ij}) with entries homogeneous forms of degree a j − b i a_j-b_i and we denote by W ( b _ ; a _ ; r ) ¯ \overline {W(\underline {b};\underline {a};r)} the closure of the locus W ( b _ ; a _ ; r ) ⊂ H i l b p ( t ) ( P n ) W(\underline {b};\underline {a};r)\subset Hilb^{p(t)}(\mathbb {P}^{n}) of determinantal schemes defined by the vanishing of the

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行列式格式的变形与无障碍

一个闭子方案X∧pn X \subset\mathbb P{^n是行列式的，如果它的齐次饱和理想可以由满足(P−s + 1) (q−s + 1) = n的齐次P × q P }\times矩阵的s × s s \times s次阵生成−dim X (p-s+1)(q-s+1)=n - \dim X，如果s= min (p,q) s= \min (p,q)，则称为标准行列式。给定整数a 1≤a 2≤⋯≤a t+c−1 a＿1 \le a＿2 \le\cdots\le a＿t+c-1{和b 1≤b 2≤⋯≤b t b＿1 }\le b＿2 \le\cdots\le b＿t我们考虑t × (t+c−1)t \times (t+c-1)矩阵A=(f ij) \mathcal A{=}(f＿ij){具有次为A j−b i a＿j-b＿i的元素齐次形式，我们用W (b)表示＿;A ＿;}r)¯\overline W({b\underline;{a;r)轨迹W (b ＿;A ＿;H i l }b p \underline(t) (pn){ W(b;a;r) }}\underline{}\underline{}\subset Hilb^{p(t)(}\mathbb p{ ^n)由

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