Projective varieties of maximal sectional regularity

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• 作者：Markus Brodmann^a ; ^{brodmann@math.unizh.ch" class="auth_mail" title="E-mail the corresponding author} ; Wanseok Lee^b ; ^{wslee@pknu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Euisung Park^c ; ^{euisungpark@korea.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Peter Schenzel^d ; ^{schenzel@informatik.uni-halle.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Primary ; 14H45 ; 13D02
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：221
• 期：1
• 页码：98-118
• 全文大小：497 K

文摘

We study projective varieties 916300706&_mathId=si1.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=186d04a5acf9b344946e8eb1136d79ac" title="Click to view the MathML source">X⊂P^r$X\subset {\mathbb{P}}^r$ of dimension 916300706&_mathId=si2.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=51156c24f315c0add5994880b6edbf29" title="Click to view the MathML source">n≥2$n\ge 2$, of codimension 916300706&_mathId=si169.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=c77a2bbee5b37acdc8f4d76dc650fb71" title="Click to view the MathML source">c≥3$c\ge 3$ and of degree 916300706&_mathId=si170.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=17f3e2590299327222b84f503ee74558" title="Click to view the MathML source">d≥c+3$d\ge c+3$ that are of maximal sectional regularity, i.e. varieties for which the Castelnuovo–Mumford regularity 916300706&_mathId=si5.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=fdc2bb7cb5b20b45e03a8358541ba191" title="Click to view the MathML source">reg(C)$\mathrm{reg}(\mathcal{C})$ of a general linear curve section is equal to 916300706&_mathId=si569.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=bc79c278c419d1f3f769823959eb4cc4" title="Click to view the MathML source">d−c+1$d-c+1$, the maximal possible value (see [10]). As one of the main results we classify all varieties of maximal sectional regularity. If X   is a variety of maximal sectional regularity, then either (a) it is a divisor on a rational normal 916300706&_mathId=si572.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=521f634294f30d92729addce6d58e915" title="Click to view the MathML source">(n+1)$(n+1)$-fold scroll 916300706&_mathId=si8.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=d99e6818d34824aca25f3d06da1a85f7" title="Click to view the MathML source">Y⊂Pⁿ⁺³$Y\subset {\mathbb{P}}^{n+3}$ or else (b) there is an n  -dimensional linear subspace 90" class="mathmlsrc">916300706&_mathId=si290.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=e9134c536eb2702627e07ee91e9be614" title="Click to view the MathML source">F⊂P^r such that 916300706&_mathId=si10.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=bcd014f8fdfc4d09443ae53fef6da26e" title="Click to view the MathML source">X∩F⊂F$X\cap \mathbb{F}\subset \mathbb{F}$ is a hypersurface of degree 916300706&_mathId=si569.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=bc79c278c419d1f3f769823959eb4cc4" title="Click to view the MathML source">d−c+1$d-c+1$. Moreover, suppose that 916300706&_mathId=si11.gif&_user=111111111&_pii=S0022404916300706&_rdoc=1&_issn=00224049&md5=d3013ed5eae3a77fc375f40bb2a56107" title="Click to view the MathML source">n=2$n=2$ or the characteristic of the ground field is zero. Then in case (b) we obtain a precise description of X as a birational linear projection of a rational normal n-fold scroll.