模手册

章节摘录

版权页：   插图：   The max in the display is achieved for ｜r-s｜≤1. Thus M naive G,{μ},C is not flat for ｜r-s｜>1, as its generic and special fibers have different dimension. We note that the analogous argument given in the proof of [75, Prop. 3.8(b)] should be amended to use the reduced special fiber in place of the honest special fiber. As always, one remedies for non-flatness of the naive local model by defining the honest local model M loc G,{μ),C to be the scheme-theoretic closure in M naive G,{μ},C of its generic fiber. Although less is known about Mloc G,{μ},C for ramified GUn than for ramified ResF/F0 GLn and ReSF/F0 GSp2g, there are by now a number of results that have been obtained in various special cases. In low rank, the case rt = 3 has been completely worked out. Theorem 2.24 ([75, 4.5, 4.15], [80, 6]). Let n=3 and (r, s)=(2,1). (i) Let C be the homothety class of the lattice A0=On F Fn. Then M naive G,{μ},C=M locG,{μ},Cthat is, Mnaive G,{μ},C is flat over Spec OF. Moreover, Mnaive G,{μ},C is normal and Cohen-Macaulay,it is smooth outside a single point y in its special fiber, and its special fiber is integral andnormal and has a rational singularity at y. The blowup Mloc G,{μ},C →Mloc G,{μ},C at y isregular with special fiber a reduced union of two smooth surfaces meeting transversely along a smooth curve. (ii) Let,C=[A1, A2], the lattice chain consisting of the homothety classes of A1 and A2. Then.Mloc G,{μ},C is smooth over Spec OF with geometric special fiber isomorphic to P2.(iii) Let,C be the standard maximal lattice chain in F3. Then Mloc G,{μ},C is normal andCohen-Macaulay. Its special fiber is reduced and consists of two irreducible components,each normal and with only rational singularities, which meet along two smooth curves which, in turn, intersect transversally at a point.

内容概要

作者：（德国）法卡斯（Gavril Farkas） （美国）莫里森（Ian Morrison）

书籍目录

Volume Ⅰ Preface Gavril Farkas and Ian Morrison Logarithmic geometry and moduli Dan Abramovich, Qile Chen, Danny Cillam, Yuhao Huang, Martin Olsson, Matthew Satriano and Shenghao Sun Invariant Hilbert schemes Michel Brion Algebraic and tropical curves:comparing their moduli spaces Lucia Caporaso A superfiaal working guide to deformations and moduli F. Catanese Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes Norman Do Equivariant geometry and the cohomology of the moduli space of curves Dan Edidin Tautological and non-tautological cohomology of the moduli space of curves C. Faber and R. Pandharipande Alternate compaaifications of moduli spaces of curves Maksym Fedorchuk and David Ishii Smyth The cohomology of the moduli space ofAbelian varieties Gerard van der Geer Moduli of K3 surfaces and irreduable symplectic manifolds V Gritsenko, K. Hulek and C.K. Sankaran Normal functions and the geometry of moduli spaces of curves Richard Hain Volume Ⅱ Parameter spaces of curves Joe Harris Global topology of the Hitchin system Tamas Hausel Differential forms on singular spaces, the minimal model program, and hyperboliaty of moduli stacks Stefan Kebekus Contractible extremal rays on Mo,n Sean Keel and James McKernan Moduli of varieties of general type Janos Kollar Singularities of stable varieties Sandor J Kovacs Soliton equations and the Riemann-Schottky problem I. Krichever and T. Shiota GIT and moduli with a twist Radu Laza Good degenerations of moduli spaces Jun Li Localization in Gromov-Witten theory and Orbifold Gromov-Witten theory Chiu-Chu Melissa Liu From WZW models to modular functors Eduard Looijenga Shimura varieties and moduli J.s. Milne The Torelli locus and special subvarieties Ben Moonen and Frans Oort Volume Ⅲ  Birational geometry for nilpotent orbits Yoshinori Namikawa Cell decompositions of moduli space, lattice points and Hurwitz problems Paul Norbury Moduli of abelian varieties in mixed and in positive characteristic Frans Oort Local models of Shimura varieties, I. Geometry and combinatorics Georgios Pappas, Michael Rapoport and Brian Smithling Generalized theta linear series on moduli spaces of vector bundles on curves Mihnea Popa Computer aided unirationality proofs of moduli spaces Frank-Olaf Schreyer Deformation theory from the point of view of fibered categories Mattia Talpo and Angelo Vistoli Mumford's conjecture-a topological outlook Ulrike Tillmann Rational parametrizations of moduli spaces of curves Alessandro Verra Hodge loci Claire Voisin Homological stability for mapping class groups of surfaces Nathalie Wahl

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《模手册(卷3)(英文版)》是由50多位活跃在代数几何领域的世界知名专家撰写的综述性文章组成。每一篇文章针对一个专题，作者力求将第一手、最新鲜的材料呈现给读者，通过介绍该专题中基础知识、例子和结论，带领读者快速进入该领域，并了解领域内重要问题；同时介绍最新的进展，使得读者能够很快捕捉到该领域最主要的文献。

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