Exceptional collections, and the Néron-Severi lattice for surfaces

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• 作者：Charles Vial¹ ; ^{c.vial@dpmms.cam.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14F05 ; 14C20 ; 14J26 ; 14J29 ; 14G27 ; 14C15
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：895-934
• 全文大小：721 K

文摘

We work out properties of smooth projective varieties X over a (not necessarily algebraically closed) field k   that admit collections of objects in the bounded derived category of coherent sheaves 3378&_mathId=si1.gif&_user=111111111&_pii=S0001870816313378&_rdoc=1&_issn=00018708&md5=034b020b998965f15d99923b1fe40b9b" title="Click to view the MathML source">D^b(X) that are either full exceptional, or numerically exceptional of maximal length. Our main result gives a necessary and sufficient condition on the Néron–Severi lattice for a smooth projective surface S   with 3378&_mathId=si132.gif&_user=111111111&_pii=S0001870816313378&_rdoc=1&_issn=00018708&md5=44489f56b7cd730bbd7b860c516b7c73" title="Click to view the MathML source">χ(O_S)=1 to admit a numerically exceptional collection of maximal length, consisting of line-bundles. As a consequence we determine exactly which complex surfaces with 3378&_mathId=si119.gif&_user=111111111&_pii=S0001870816313378&_rdoc=1&_issn=00018708&md5=fa5cb2b09cbdeee9d40921e8ce702034" title="Click to view the MathML source">p_g=q=0 admit a numerically exceptional collection of maximal length. Another consequence is that a minimal geometrically rational surface with a numerically exceptional collection of maximal length is rational.