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Books on the topic 'Algebraic intersection'

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Published: 4 May 2024

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1

Armand, Borel, ed. Intersection cohomology. Boston: Birkhäuser, 2008.

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2

Ellingsrud, Geir. Recent Progress in Intersection Theory. Boston, MA: Birkhäuser Boston, 2000.

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3

1948-, Masser David William, ed. Some problems of unlikely intersections in arithmetic and geometry. Princeton: Princeton University Press, 2012.

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4

Cho, Yŏng-hyŏn. Taesu kihahak: Wanjŏn kyochʻa rŭl chungsim ŭro. Sŏul: Minŭmsa, 1991.

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5

Mandal, Satya. Projective modules and complete intersections. Berlin: Springer, 1997.

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6

Rumely, Robert S. Capacity theory on algebraic curves. Berlin: Springer-Verlag, 1989.

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7

Zeuthen Symposium (1989 Mathematical Institute of the University of Copenhagen). Enumerative algebraic geometry: Proceedings of the 1989 Zeuthen Symposium. Edited by Zeuthen H. G. 1839-1920, Kleiman Steven L, Thorup Anders 1943-, and Statens naturvidenskabelige forskningsråd (Denmark). Providence, R.I: American Mathematical Society, 1991.

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8

Uwe, Storch, ed. Regular sequences and resultants. Natick, Mass: A.K. Peters, 2001.

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9

G, Rodicio Antonio, ed. Smoothness, regularity and complete intersection. Cambridge: Cambridge University Press, 2010.

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10

Kirwan, Frances. An introductionto intersection homology theory. Harlow: Longman Scientific & Technical, 1988.

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11

Dias, Danielle. Configuration spaces over Hilbert schemes and applications. Berlin: Springer, 1996.

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12

Intersections de deux quadriques et pinceaux de courbes de genre 1 =: Intersections of two quadrics and pencils of curves of genus 1. Berlin: Springer, 2007.

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13

Ebeling, Wolfgang. The monodromy groups of isolated singularities of complete intersections. Berlin: Springer-Verlag, 1987.

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14

Grattan-Guinness, I. Algebras, projective geometry, mathematical logic and constructing the world: Intersections in the philosophy of mathematics of A.N.Whitehead. London: Middlesex University Business School, 2002.

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15

Pan-American Advanced Studies Institute (2009 Universidade Federal de Pernambuco). Commutative algebra and its connections to geometry: Pan-American Advanced Studies Institute, August 3--14, 2009, Universidade Federal de Pernambuco, Olinda, Brazil. Edited by Corso Alberto and Polini Claudia 1966-. Providence, R.I: American Mathematical Society, 2011.

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16

Erds-Ko-Rado Theorems: Algebraic Approaches. Cambridge University Press, 2015.

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17

Godsil, Christopher, and Karen Meagher. Erds-Ko-Rado Theorems: Algebraic Approaches. Cambridge University Press, 2015.

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18

Flenner, H. Joins and Intersections. Springer, 2010.

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19

Flenner, H., L. O'Carroll, and W. Vogel. Joins and Intersections. Springer, 2013.

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20

Flenner, H., L. O'Carroll, and W. Vogel. Joins and Intersections (Springer Monographs in Mathematics). Springer, 1999.

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21

Segre's Reflexivity and an Inductive Characterization of Hyperquadrics (Memoirs of the American Mathematical Society, No. 763). American Mathematical Society, 2002.

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22

Rumely, Robert S. Capacity Theory on Algebraic Curves. Springer, 1989.

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23

Rumely, Robert S. Capacity Theory on Algebraic Curves. Springer London, Limited, 2006.

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24

Pedersen, Jean, and Hans Walser. 99 Points of Intersection: Examples-Pictures-Proofs. The Mathematical Association of America, 2006.

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25

Aluffi, Paolo, David Anderson, Milena Hering, Mircea Mustaţă, and Sam Payne, eds. Facets of Algebraic Geometry. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108877831.

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Written to honor the 80th birthday of William Fulton, the articles collected in this volume (the first of a pair) present substantial contributions to algebraic geometry and related fields, with an emphasis on combinatorial algebraic geometry and intersection theory. Featured topics include commutative algebra, moduli spaces, quantum cohomology, representation theory, Schubert calculus, and toric and tropical geometry. The range of these contributions is a testament to the breadth and depth of Fulton's mathematical influence. The authors are all internationally recognized experts, and include well-established researchers as well as rising stars of a new generation of mathematicians. The text aims to stimulate progress and provide inspiration to graduate students and researchers in the field.

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26

Aluffi, Paolo, David Anderson, Milena Hering, Mircea Mustaţă, and Sam Payne, eds. Facets of Algebraic Geometry. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108877855.

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Written to honor the 80th birthday of William Fulton, the articles collected in this volume (the second of a pair) present substantial contributions to algebraic geometry and related fields, with an emphasis on combinatorial algebraic geometry and intersection theory. Featured include commutative algebra, moduli spaces, quantum cohomology, representation theory, Schubert calculus, and toric and tropical geometry. The range of these contributions is a testament to the breadth and depth of Fulton's mathematical influence. The authors are all internationally recognized experts, and include well-established researchers as well as rising stars of a new generation of mathematicians. The text aims to stimulate progress and provide inspiration to graduate students and researchers in the field.

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27

(Editor), Jan O. Kleppe, Juan C. Migliore (Editor), and Rosa Miro-Roig (Editor), eds. Gorenstein Liaison, Complete Intersection Liaison Invariants and. American Mathematical Society, 2001.

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28

El Karoui, Noureddine. Algebraic geometry and matrix models. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.29.

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This article discusses the connection between the matrix models and algebraic geometry. In particular, it considers three specific applications of matrix models to algebraic geometry, namely: the Kontsevich matrix model that describes intersection indices on moduli spaces of curves with marked points; the Hermitian matrix model free energy at the leading expansion order as the prepotential of the Seiberg-Witten-Whitham-Krichever hierarchy; and the other orders of free energy and resolvent expansions as symplectic invariants and possibly amplitudes of open/closed strings. The article first describes the moduli space of algebraic curves and its parameterization via the Jenkins-Strebel differentials before analysing the relation between the so-called formal matrix models (solutions of the loop equation) and algebraic hierarchies of Dijkgraaf-Witten-Whitham-Krichever type. It also presents the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations, along with higher expansion terms and symplectic invariants.

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29

Green, Mark L. The Spread Philosophy in the Study of Algebraic Cycles. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0010.

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This chapter discusses the spread philosophy in the study of algebraic cycles, in order to make use of a geometry by considering a variation of Hodge structure where D is the Hodge domain (or the appropriate Mumford–Tate domain) and Γ‎ is the group of automorphisms of the integral lattice preserving the intersection pairing. If we have an algebraic cycle Z on X, taking spreads yields a cycle Ƶ on X. Applying Hodge theory to Ƶ on X gives invariants of the cycle. Another related situation is algebraic K-theory. For example, to study Kₚsuperscript Milnor(k), the geometry of S can be used to construct invariants.

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30

Scheja, Gunter, and Uwe Storch. Regular Sequences and Resultants: Research Notes in Mathematics, Volume 8 (Research Notes in Mathematics (Boston, Mass.), 8.). AK Peters, 2001.

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31

Cataldo, Mark Andrea de, Luca Migliorini Lectures 4–5, and Mark Andrea de Cataldo. The Hodge Theory of Maps. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0006.

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This chapter showcases two further lectures on the Hodge theory of maps, and they are mostly composed of exercises. The first lecture details a minimalist approach to sheaf cohomology, and then turns to the intersection cohomology complex, which is limited to the definition and calculation of the intersection complex Isubscript X of a variety of dimension d with one isolated singularity. Finally, this lecture discusses the Verdier duality. The second lecture sets out the Decomposition theorem, which is the deepest known fact concerning the homology of algebraic varieties. It then considers the relative hard Lefschetz and the hard Lefschetz for intersection cohomology groups.

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32

Farb, Benson, and Dan Margalit. Pseudo-Anosov Theory. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691147949.003.0015.

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This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

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33

Farb, Benson, and Dan Margalit. Dehn Twists. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691147949.003.0004.

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This chapter deals with Dehn twists, the simplest infinite-order elements of Mod(S). It first defines Dehn twists and proves that they are nontrivial elements of the mapping class group. In particular, it considers the action of Dehn twists on simple closed curves. As one application of this study, the chapter proves that if two simple closed curves in Sɡ have geometric intersection number greater than 1, then the associated Dehn twists generate a free group of rank 2 in Mod(S). It also proves some fundamental facts about Dehn twists and describes the center of the mapping class group, along with algebraic relations that can occur between two Dehn twists. Finally, it explores three geometric operations on a surface that each induces an algebraic operation on the corresponding mapping class group: the inclusion homomorphism, the capping homomorphism, and the cutting homomorphism.

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34

Farb, Benson, and Dan Margalit. The Symplectic Representation and the Torelli Group. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691147949.003.0007.

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This chapter discusses the basic properties and applications of a symplectic representation, denoted by Ψ‎, and its kernel, called the Torelli group. After describing the algebraic intersection number as a symplectic form, the chapter presents three different proofs of the surjectivity of Ψ‎, each illustrating a different theme. It also illustrates the usefulness of the symplectic representation by two applications to understanding the algebraic structure of Mod(S). First, the chapter explains how this representation is used by Serre to prove the theorem that Mod(Sɡ) has a torsion-free subgroup of finite index. It thens uses the symplectic representation to prove, following Ivanov, the following theorem of Grossman: Mod(Sɡ) is residually finite. It also considers some of the pioneering work of Dennis Johnson on the Torelli group. In particular, a Johnson homomorphism is constructed and some of its applications are given.

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35

Majadas, Javier, and Antonio G. Rodicio. Smoothness, Regularity and Complete Intersection. Cambridge University Press, 2011.

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36

Majadas, Javier, and Antonio G. Rodicio. Smoothness, Regularity and Complete Intersection. Cambridge University Press, 2010.

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37

Majadas, Javier, and Antonio G. Rodicio. Smoothness, Regularity and Complete Intersection. Cambridge University Press, 2011.

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38

Majadas, Javier, and Antonio G. Rodicio. Smoothness, Regularity and Complete Intersection. Cambridge University Press, 2010.

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39

Intersections Of Hirzebruchzagier Divisors And Cm Cycles. Springer, 2012.

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40

Perverse Sheaves and Applications to Representation Theory. American Mathematical Society, 2021.

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41

Howard, Benjamin, and Tonghai Yang. Intersections of Hirzebruch-Zagier Divisors and CM Cycles. Springer, 2012.

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42

Hilbert Schemes of Points and Infinite Dimensional Lie Algebras. American Mathematical Society, 2018.

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43

Diagrammatic Algebra. American Mathematical Society, 2021.

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44

Lecture Notes on Local Rings. World Scientific, 2014.

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45

Ebeling, Wolfgang. Monodromy Groups of Isolated Singularities of Complete Intersections. Springer London, Limited, 2006.

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46

Voisin, Claire. Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187). Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691160504.001.0001.

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This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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47

Bounded Cohomology of Discrete Groups. American Mathematical Society, 2017.

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