Deformation and Unobstructedness of Determinantal Schemes

IF 2 4区数学 Q1 MATHEMATICS

Memoirs of the American Mathematical Society 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":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":2.0000,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Memoirs of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/memo/1418","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","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

查看原文本刊更多论文

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

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Memoirs of the American Mathematical Society 数学-数学

CiteScore

3.50

自引率

5.30%

发文量

审稿时长

>12 weeks

期刊介绍： Memoirs of the American Mathematical Society is devoted to the publication of research in all areas of pure and applied mathematics. The Memoirs is designed particularly to publish long papers or groups of cognate papers in book form, and is under the supervision of the Editorial Committee of the AMS journal Transactions of the AMS. To be accepted by the editorial board, manuscripts must be correct, new, and significant. Further, they must be well written and of interest to a substantial number of mathematicians.

京公网安备 11010802042870号

Book学术文献互助群
群号：481959085