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OverviewHilbert schemes, which parametrize subschemes in algebraic varieties, have been extensively studied in algebraic geometry for the last 50 years. The most interesting class of Hilbert schemes are schemes $X^{[n]}$ of collections of $n$ points (zero-dimensional subschemes) in a smooth algebraic surface $X$. Schemes $X^{[n]}$ turn out to be closely related to many areas of mathematics, such as algebraic combinatorics, integrable systems, representation theory, and mathematical physics, among others. This book surveys recent developments of the theory of Hilbert schemes of points on complex surfaces and its interplay with infinite dimensional Lie algebras. It starts with the basics of Hilbert schemes of points and presents in detail an example of Hilbert schemes of points on the projective plane. Then the author turns to the study of cohomology of $X^{[n]}$, including the construction of the action of infinite dimensional Lie algebras on this cohomology, the ring structure of cohomology, equivariant cohomology of $X^{[n]}$ and the Gromov-Witten correspondence. The last part of the book presents results about quantum cohomology of $X^{[n]}$ and related questions. The book is of interest to graduate students and researchers in algebraic geometry, representation theory, combinatorics, topology, number theory, and theoretical physics. Full Product DetailsAuthor: Zhenbo QinPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.765kg ISBN: 9781470441883ISBN 10: 1470441888 Pages: 336 Publication Date: 30 March 2018 Audience: Professional and scholarly , College/higher education , Professional & Vocational , Postgraduate, Research & Scholarly Format: Hardback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsHilbert schemes of points on surfaces: Basic results on Hilbert schemes of points The nef cone and flip structure of $(\mathbb{P}^2)^{[n]}$ Hilbert schemes and infinite dimensional Lie algebras: Hilbert schemes and infinite dimensional Lie algebras Chern character operators Multiple $q$-zeta values and Hilbert schemes Lie algebras and incidence Hilbert schemes Cohomology rings of Hilbert schemes of points: The cohomology rings of Hilbert schemes of points on surfaces Ideals of the cohomology rings of Hilbert schemes Integral cohomology of Hilbert schemes The ring structure of $H^*_{\textrm{orb}}(X^{(n)})$ Equivariant cohomology of the Hilbert schemes of points: Equivariant cohomology of Hilbert schemes Hilbert/Gromov-Witten correspondence Gromov-Witten theory of the Hilbert schemes of points: Cosection localization for the Hilbert schemes of points Equivariant quantum operator of Okounkov-Pandharipande The genus-0 extremal Gromov-Witten invariants Ruan's Cohomological Crepant Resolution Conjecture Bibliography IndexReviewsAuthor InformationZhenbo Qin, University of Missouri, Columbia, MO. Tab Content 6Author Website:Countries AvailableAll regions |
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