LEADER 03963nam 22005415 450 001 9911031640803321 005 20251003130946.0 010 $a3-031-86043-8 024 7 $a10.1007/978-3-031-86043-0 035 $a(MiAaPQ)EBC32327742 035 $a(Au-PeEL)EBL32327742 035 $a(CKB)41543259800041 035 $a(DE-He213)978-3-031-86043-0 035 $a(EXLCZ)9941543259800041 100 $a20251003d2025 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aIntroduction to Singularities and Deformations /$fby Gert-Martin Greuel, Christoph Lossen, Eugenii I. Shustin 205 $a2nd ed. 2025. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2025. 215 $a1 online resource (734 pages) 225 1 $aSpringer Monographs in Mathematics,$x2196-9922 311 08$a3-031-86042-X 327 $a1 Singularity Theory -- 2 Local Deformation Theory -- 3 Singularities in Arbitrary Characteristics -- Appendix A: Sheaves -- Appendix B: Commutative Algebra -- Appendix C: Formal Deformation Theory. 330 $aIn the second edition we do not only correct errors, update references and improve some of the proofs of the text of the first edition, but also add a new chapter on singularities in arbitrary characteristic. We give an overview of several aspects of singularities of algebraic varieties and formal power series defined over a field of arbitrary characteristic (algebraically closed or not). Almost all of the results presented here appeared after the publication of the first edition and some results are new. In particular, we treat, in arbitrary characteristic, the classical invariants of hypersurface singularities, and we review results on the equisingularity of plane curve singularities, on the classification of parametrizations of plane branches, and on hypersurface and complete intersection singularities with small moduli. Moreover, we discuss and prove determinacy and semicontinuity results of families of ideals and matrices of power series parametrized by an arbitrary Noether base scheme, which are used to prove open loci properties for several singularity invariants. The semicontinuity has surprising applications in the computation of local standard bases of zero dimensional ideals, which are by magnitudes faster than previously known methods. The chapter contains two appendices. One is by Dmitry Kerner on large submodules within group orbits, which relates to determinacy criteria for singularities in very general contexts. It is focused on methods applicable to a broad class of fields of arbitrary characteristic, while before the theory was mainly restricted to zero characteristic. The second appendix is by Ilya Tyomkin and deals with the geometry of Severi varieties, mainly on toric varieties. It discusses the breakthrough solution to the problem on the irreducibility of Severi varieties of the plane in arbitrary characteristic, with a focus on the characteristic free approach based on tropical geometry. We try to be self-contained and give proofs whenever possible. However, due to the amount of material, this is not always possible, and we then give precise references to the original sources. 410 0$aSpringer Monographs in Mathematics,$x2196-9922 606 $aAlgebra 606 $aGeometry, Algebraic 606 $aAlgebra 606 $aAlgebraic Geometry 615 0$aAlgebra. 615 0$aGeometry, Algebraic. 615 14$aAlgebra. 615 24$aAlgebraic Geometry. 676 $a515/.942 700 $aGreuel$b Gert-Martin$0767897 701 $aLossen$b Christoph$0767898 701 $aShustin$b Eugenii I$01851503 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9911031640803321 996 $aIntroduction to Singularities and Deformations$94445297 997 $aUNINA