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Hardcover ISBN:  9781470441883 
Product Code:  SURV/228 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470443894 
Product Code:  SURV/228.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470441883 
eBook ISBN:  9781470443894 
Product Code:  SURV/228.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 228; 2018; 336 ppMSC: Primary 14; 17
Hilbert 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 (zerodimensional 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.
ReadershipGraduate students and researchers interested in algebraic geometry.

Table of Contents

Hilbert 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/GromovWitten correspondence

GromovWitten theory of the Hilbert schemes of points

Cosection localization for the Hilbert schemes of points

Equivariant quantum operator of OkounkovPandharipande

The genus0 extremal GromovWitten invariants

Ruan’s Cohomological Crepant Resolution Conjecture


Additional Material

Reviews

The book is far from elementary and is suitable for researchers and graduate students with a good knowledge of intersection theory and geometric invariant theory. Starting with this knowledge, it surveys most of the known results and theories of the Hilbert schemes of points, implying infinite dimensional Lie Algebras and their actions. Most of the contemporary theory in the field is included in the book, an impressive bibliography is given, and all results are proved in full detail...This makes the book complete, and makes it a splendid survey, even textbook, on the subject.
Arvid Siqveland, Zentralblatt MATH


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Hilbert 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 (zerodimensional 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.
Graduate students and researchers interested in algebraic geometry.

Hilbert 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/GromovWitten correspondence

GromovWitten theory of the Hilbert schemes of points

Cosection localization for the Hilbert schemes of points

Equivariant quantum operator of OkounkovPandharipande

The genus0 extremal GromovWitten invariants

Ruan’s Cohomological Crepant Resolution Conjecture

The book is far from elementary and is suitable for researchers and graduate students with a good knowledge of intersection theory and geometric invariant theory. Starting with this knowledge, it surveys most of the known results and theories of the Hilbert schemes of points, implying infinite dimensional Lie Algebras and their actions. Most of the contemporary theory in the field is included in the book, an impressive bibliography is given, and all results are proved in full detail...This makes the book complete, and makes it a splendid survey, even textbook, on the subject.
Arvid Siqveland, Zentralblatt MATH