Cremona groups and the icosahedron /
Cremona Groups and the Icosahedron focuses on the Cremona groups of ranks 2 and 3 and describes the beautiful appearances of the icosahedral group A5 in them. The book surveys known facts about surfaces with an action of A5, explores A5-equivariant geometry of the quintic del Pezzo threefold V5, and...
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| Format: | eBook |
| Language: | English |
| Published: |
Boca Raton :
Chapman and Hall/CRC,
2015.
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| Series: | Monographs and research notes in mathematics.
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| Subjects: | |
| Online Access: | Connect to the full text of this electronic book |
| Summary: | Cremona Groups and the Icosahedron focuses on the Cremona groups of ranks 2 and 3 and describes the beautiful appearances of the icosahedral group A5 in them. The book surveys known facts about surfaces with an action of A5, explores A5-equivariant geometry of the quintic del Pezzo threefold V5, and gives a proof of its A5-birational rigidity. The authors explicitly describe many interesting A5-invariant subvarieties of V5, including A5-orbits, low-degree curves, invariant anticanonical K3 surfaces, and a mildly singular surface of general type that is a degree five cover of the diagonal Clebsch cubic surface. They also present two birational selfmaps of V5 that commute with A5-action and use them to determine the whole group of A5-birational automorphisms. As a result of this study, they produce three non-conjugate icosahedral subgroups in the Cremona group of rank 3, one of them arising from the threefold V5. This book presents up-to-date tools for studying birational geometry of higher-dimensional varieties. In particular, it provides readers with a deep understanding of the biregular and birational geometry of V5. |
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| Item Description: | <P><strong>Introduction <br></strong>Conjugacy in Cremona groups <br>Three-dimensional projective space <br>Other rational Fano threefolds <br>Statement of the main result <br>Outline of the book </p> <p><b><i> <p>Preliminaries <br></i>Singularities of pairs</b><br>Canonical and log canonical singularities <br>Log pairs with mobile boundaries <br>Multiplier ideal sheaves <br>Centers of log canonical singularities <br>Corti's inequality </p> <p><b> <p>Noether-Fano inequalities</b> <br>Birational rigidity <br>Fano varieties and elliptic fibrations <br>Applications to birational rigidity <br>Halphen pencils </p> <p><b> <p>Auxiliary results <br></b>Zero-dimensional subschemes <br>Atiyah flops <br>One-dimensional linear systems <br>Miscellanea </p> <p><b><i> <p>Icosahedral Group <br></i>Basic properties</b> <br>Action on points and curves <br>Representation theory <br>Invariant theory <br>Curves of low genera <br>SL<sub>2</sub>(C) and PSL<sub>2</sub>(C) <br>Binary icosahedral group <br>Symmetric group <br>Dihedral group </p> <p><b> <p>Surfaces with icosahedral symmetry</b> <br>Projective plane <br>Quintic del Pezzo surface <br>Clebsch cubic surface <br>Two-dimensional quadric <br>Hirzebruch surfaces <br>Icosahedral subgroups of Cr<sub>2</sub>(C) <br><i>K</i>3 surfaces </p> <p><b><i> <p>Quintic del Pezzo Threefold</i> <br>Quintic del Pezzo threefold <br></b>Construction and basic properties <br>PSL<sub>2</sub>(C)-invariant anticanonical surface <br>Small orbits <br>Lines <br>Orbit of length five <br>Five hyperplane sections <br>Projection from a line <br>Conics </p> <p><b> <p>Anticanonical linear system</b> <br>Invariant anticanonical surfaces <br>Singularities of invariant anticanonical surfaces <br>Curves in invariant anticanonical surfaces </p> <p><b> <p>Combinatorics of lines and conics</b> <br>Lines <br>Conics </p> <p><b> <p>Special invariant curves</b><br>Irreducible curves <br>Preliminary classification of low degree curves </p> <p><b> <p>Two Sarkisov links</b> <br>Anticanonical divisors through the curve L<sub>6</sub> <br>Rational map to P<sup>4</sup> <br>A remarkable sextic curve <br>Two Sarkisov links <br>Action on the Picard group </p> <p><b><i> <p>Invariant Subvarieties</i> <br>Invariant cubic hypersurface<br></b>Linear system of cubics <br>Curves in the invariant cubic <br>Bring's curve in the invariant cubic <br>Intersecting invariant quadrics and cubic <br>A remarkable rational surface </p> <p><b> <p>Curves of low degree <br></b>Curves of degree 16 <br>Six twisted cubics <br>Irreducible curves of degree 18 <br>A singular curve of degree 18 <br>Bring's curve <br>Classification </p> <p><b> <p>Orbits of small length</b><br>Orbits of length 20 <br>Ten conics <br>Orbits of length 30 <br>Fifteen twisted cubics </p> <p><b> <p>Further properties of the invariant cubic</b> <br>Intersections with low degree curves <br>Singularities of the invariant cubic <br>Projection to Clebsch cubic surface <br>Picard group </p> <p><b> <p>Summary of orbits, curves, and surfaces</b> <br>Orbits vs. curves <br>Orbits vs. surfaces <br>Curves vs. surfaces <br>Curves vs. curves </p> <p><b><i> <p>Singularities of Linear Systems</i></b> <br><b>Base loci of invariant linear systems</b> <br>Orbits of length 10 <br>Linear system Q<sub>3</sub> <br>Isolation of orbits in S <br>Isolation of arbitrary orbits <br>Isolation of the curve L<sub>15</sub> </p> <p><b> <p>Proof of the main result</b> <br>Singularities of linear systems <br>Restricting divisors to invariant quadrics <br>Exclusion of points and curves different from L<sub>15</sub> <br>Exclusion of the curve L<sub>15</sub> <br>Alternative approach to exclusion of points <br>Alternative approach to the exclusion of L<sub>15</sub> </p> <p><b> <p>Halphen pencils and elliptic fibrations</b> <br>Statement of results <br>Exclusion of points <br>Exclusion of curves <br>Description of non-terminal pairs <br>Completing the proof</p> |
| Physical Description: | 1 online resource (xxi, 504 pages) : illustrations (black and white) |
| Bibliography: | Includes bibliographical references and index. |
| ISBN: | 9781482251609 1482251604 9781482251593 1482251590 |