LEADER 05333nam 22007095 450 001 996466485303316 005 20200707032832.0 010 $a3-540-44646-X 024 7 $a10.1007/b100262 035 $a(CKB)1000000000231538 035 $a(SSID)ssj0000327520 035 $a(PQKBManifestationID)11268708 035 $a(PQKBTitleCode)TC0000327520 035 $a(PQKBWorkID)10301750 035 $a(PQKB)11322681 035 $a(DE-He213)978-3-540-44646-0 035 $a(MiAaPQ)EBC3087322 035 $a(PPN)155177389 035 $a(EXLCZ)991000000000231538 100 $a20121227d2004 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 14$aThe Valuative Tree$b[electronic resource] /$fby Charles Favre, Mattias Jonsson 205 $a1st ed. 2004. 210 1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d2004. 215 $a1 online resource (XVI, 244 p.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v1853 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-540-22984-1 320 $aIncludes bibliographical references and index. 327 $a1 Generalities -- 1.1 Setup -- 1.2 Valuations -- 1.3 Krull Valuations -- 1.4 Plane Curves -- 1.5 Examples of Valuations -- 1.6 Valuations Versus Krull Valuations -- 1.7 Sequences of Blowups and Krull Valuations -- 2 MacLane?s Method -- 2.1 Sequences of Key Polynomials -- 2.2 Classification -- 2.3 Graded Rings and Numerical Invariants -- 2.4 From Valuations to SKP?s -- 2.5 A Computation -- 3 Tree Structures -- 3.1 Trees -- 3.2 Nonmetric Tree Structure on V -- 3.3 Parameterization of V by Skewness -- 3.4 Multiplicities -- 3.5 Approximating Sequences -- 3.6 Thinness -- 3.7 Value Semigroups and Approximating Sequences -- 3.8 Balls of Curves -- 3.9 The Relative Tree Structure -- 4 Valuations Through Puiseux Series -- 4.1 Puiseux Series and Valuations -- 4.2 Tree Structure -- 4.3 Galois Action -- 4.4 A Tale of Two Trees -- 4.5 The Berkovich Projective Line -- 4.6 The Bruhat-Tits Metric -- 4.7 Dictionary -- 5 Topologies -- 5.1 The Weak Topology -- 5.2 The Strong Topology on V -- 5.3 The Strong Topology on Vqm -- 5.4 Thin Topologies -- 5.5 The Zariski Topology -- 5.6 The Hausdorff-Zariski Topology -- 5.7 Comparison of Topologies -- 6 The Universal Dual Graph -- 6.1 Nonmetric Tree Structure -- 6.2 Infinitely Near Points -- 6.3 Parameterization and Multiplicity -- 6.4 The Isomorphism -- 6.5 Proof of the Isomorphism -- 6.6 Applications -- 6.7 The Dual Graph of the Minimal Desingularization -- 6.8 The Relative Tree Structure -- 7 Tree Measures -- 7.1 Outline -- 7.2 More on the Weak Topology -- 7.3 Borel Measures -- 7.4 Functions of Bounded Variation -- 7.5 Representation Theorem I -- 7.6 Complex Tree Potentials -- 7.7 Representation Theorem II -- 7.8 Atomic Measures -- 7.9 Positive Tree Potentials -- 7.10 Weak Topologies and Compactness -- 7.11 Restrictions to Subtrees -- 7.12 Inner Products -- 8 Applications of the Tree Analysis -- 8.1 Zariski?s Theory of Complete Ideals -- 8.2 The Voûte étoilée -- A Infinitely Singular Valuations -- A.1 Characterizations -- A.2 Constructions -- B The Tangent Space at a Divisorial Valuation -- C Classification -- D Combinatorics of Plane Curve Singularities -- D.1 Zariski?s Terminology for Plane Curve Singularities -- D.2 The Eggers Tree -- E.1 Completeness -- E.2 The Residue Field -- References. 330 $aThis volume is devoted to a beautiful object, called the valuative tree and designed as a powerful tool for the study of singularities in two complex dimensions. Its intricate yet manageable structure can be analyzed by both algebraic and geometric means. Many types of singularities, including those of curves, ideals, and plurisubharmonic functions, can be encoded in terms of positive measures on the valuative tree. The construction of these measures uses a natural tree Laplace operator of independent interest. 410 0$aLecture Notes in Mathematics,$x0075-8434 ;$v1853 606 $aDiscrete mathematics 606 $aAlgebraic geometry 606 $aCommutative algebra 606 $aCommutative rings 606 $aTopology 606 $aDiscrete Mathematics$3https://scigraph.springernature.com/ontologies/product-market-codes/M29000 606 $aAlgebraic Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M11019 606 $aCommutative Rings and Algebras$3https://scigraph.springernature.com/ontologies/product-market-codes/M11043 606 $aTopology$3https://scigraph.springernature.com/ontologies/product-market-codes/M28000 615 0$aDiscrete mathematics. 615 0$aAlgebraic geometry. 615 0$aCommutative algebra. 615 0$aCommutative rings. 615 0$aTopology. 615 14$aDiscrete Mathematics. 615 24$aAlgebraic Geometry. 615 24$aCommutative Rings and Algebras. 615 24$aTopology. 676 $a510 s 676 $a511/.52 700 $aFavre$b Charles$4aut$4http://id.loc.gov/vocabulary/relators/aut$0478896 702 $aJonsson$b Mattias$4aut$4http://id.loc.gov/vocabulary/relators/aut 906 $aBOOK 912 $a996466485303316 996 $aThe Valuative Tree$92530052 997 $aUNISA