Artin fans. AMS special session on Combinatorics and Algebraic Geometry. Dan Abramovich. Brown University. October 24, 2014
|
|
|
- Jennifer Cummings
- 5 years ago
- Views:
Transcription
1 Artin fans AMS special session on Combinatorics and Algebraic Geometry Dan Abramovich Brown University October 24, 2014 Abramovich (Brown) Artin fans October 24, / 13
2 Heros: Abramovich (Brown) Artin fans October 24, / 13
3 Heros: Martin Olsson Abramovich (Brown) Artin fans October 24, / 13
4 Heros: Martin Olsson Jonathan Wise Qile Chen, Steffen Marcus, Mark Gross, Bernd Siebert Abramovich (Brown) Artin fans October 24, / 13
5 Heros: Martin Olsson Jonathan Wise Qile Chen, Steffen Marcus, Mark Gross, Bernd Siebert Martin Ulirsch Abramovich (Brown) Artin fans October 24, / 13
6 Superabundance Mikhalkin-Speyer: there is a tropical cubic curve C of genus 1 in TP 3 which does not lift to an algebraic curve (Speyer, Tropical Geometry, Berkeley thesis 2005, Figure 5.1). Abramovich (Brown) Artin fans October 24, / 13
7 Superabundance (continued) I want to understand this phenomenon. Abramovich (Brown) Artin fans October 24, / 13
8 Superabundance (continued) I want to understand this phenomenon. Principles: Abramovich (Brown) Artin fans October 24, / 13
9 Superabundance (continued) I want to understand this phenomenon. Principles: Tropical curves in TP 3 encode degenerations of curves in P 3 Abramovich (Brown) Artin fans October 24, / 13
10 Superabundance (continued) I want to understand this phenomenon. Principles: Tropical curves in TP 3 encode degenerations of curves in P 3 They encode in detail the manner in which they degenerate Abramovich (Brown) Artin fans October 24, / 13
11 Superabundance (continued) I want to understand this phenomenon. Principles: Tropical curves in TP 3 encode degenerations of curves in P 3 They encode in detail the manner in which they degenerate They encode logarithmic stable maps in P 3. Abramovich (Brown) Artin fans October 24, / 13
12 Superabundance (continued) I want to understand this phenomenon. Principles: Tropical curves in TP 3 encode degenerations of curves in P 3 They encode in detail the manner in which they degenerate They encode logarithmic stable maps in P 3. But logarithmic stable maps are obstructed. Abramovich (Brown) Artin fans October 24, / 13
13 Superabundance (continued) I want to understand this phenomenon. Principles: Tropical curves in TP 3 encode degenerations of curves in P 3 They encode in detail the manner in which they degenerate They encode logarithmic stable maps in P 3. But logarithmic stable maps are obstructed. Question Is there a world in which they are not obstructed? Abramovich (Brown) Artin fans October 24, / 13
14 Logarithmic structures Definition A pre logarithmic structure is X = (X, M α O X ) Abramovich (Brown) Artin fans October 24, / 13
15 Logarithmic structures Definition A pre logarithmic structure is X = (X, M α O X ) or just (X, M) Abramovich (Brown) Artin fans October 24, / 13
16 Logarithmic structures Definition A pre logarithmic structure is such that X = (X, M α O X ) or just (X, M) X is a scheme - the underlying scheme Abramovich (Brown) Artin fans October 24, / 13
17 Logarithmic structures Definition A pre logarithmic structure is such that X = (X, M α O X ) or just (X, M) X is a scheme - the underlying scheme M is a sheaf of monoids on X, and Abramovich (Brown) Artin fans October 24, / 13
18 Logarithmic structures Definition A pre logarithmic structure is such that X = (X, M α O X ) or just (X, M) X is a scheme - the underlying scheme M is a sheaf of monoids on X, and α is a monoid homomorphism, where the monoid structure on O X is the multiplicative structure. Abramovich (Brown) Artin fans October 24, / 13
19 Logarithmic structures Definition A pre logarithmic structure is such that X = (X, M α O X ) or just (X, M) X is a scheme - the underlying scheme M is a sheaf of monoids on X, and α is a monoid homomorphism, where the monoid structure on O X is the multiplicative structure. Definition It is a logarithmic structure if α : α 1 O X O X is an isomorphism. Abramovich (Brown) Artin fans October 24, / 13
20 Examples Examples (X, O X O X ), the trivial logarithmic structure. Abramovich (Brown) Artin fans October 24, / 13
21 Examples Examples (X, O X O X ), the trivial logarithmic structure. Let X, D X be a variety with a divisor. We define M D O X : Abramovich (Brown) Artin fans October 24, / 13
22 Examples Examples (X, O X O X ), the trivial logarithmic structure. Let X, D X be a variety with a divisor. We define M D O X : M D (U) = {f O X (U) f UD O X (U D) }. Abramovich (Brown) Artin fans October 24, / 13
23 Examples Examples (X, O X O X ), the trivial logarithmic structure. Let X, D X be a variety with a divisor. We define M D O X : M D (U) = {f O X (U) f UD O X (U D) }. Let k be a field, N k k (n, z) z 0 n defined by sending 0 1 and n 0 otherwise. Abramovich (Brown) Artin fans October 24, / 13
24 The magic of logarithmic geomery Any toric variety X is logarithmically smooth T X O dim X X. Abramovich (Brown) Artin fans October 24, / 13
25 The magic of logarithmic geomery Any toric variety X is logarithmically smooth T X O dim X X. A nodal curve is logarithmically smooth over a logarithmic point. Abramovich (Brown) Artin fans October 24, / 13
26 Here be monsters! Logarithmic obstructions to deforming a logarithmic map C P 3 lie in H 1 (C, O 3 C ). Abramovich (Brown) Artin fans October 24, / 13
27 Here be monsters! Logarithmic obstructions to deforming a logarithmic map C P 3 lie in H 1 (C, O 3 C ). These can be nonzero on a broken cubic curve! Abramovich (Brown) Artin fans October 24, / 13
28 Artin fans Olsson: {Logarithmic structures X on X } {X Log}. Abramovich (Brown) Artin fans October 24, / 13
29 Artin fans Olsson: {Logarithmic structures X on X } {X Log}. The stack Log is huge and does not specify combinatorial data. Abramovich (Brown) Artin fans October 24, / 13
30 Artin fans Olsson: {Logarithmic structures X on X } {X Log}. The stack Log is huge and does not specify combinatorial data. Proposition (Wise; ℵ, Chen, Marcus) There is an initial factorization X A X Log such that A X Log is étale, representable, strict. Abramovich (Brown) Artin fans October 24, / 13
31 Artin fans Olsson: {Logarithmic structures X on X } {X Log}. The stack Log is huge and does not specify combinatorial data. Proposition (Wise; ℵ, Chen, Marcus) There is an initial factorization X A X Log such that A X Log is étale, representable, strict. The stack A X is small, totally combinatorial. Abramovich (Brown) Artin fans October 24, / 13
32 P 3 and A P 3 P 3 = (A 4 {0})/G m. So {C P 3 } {(L, s 0,..., s 3 ) s i do not vanish together}. Abramovich (Brown) Artin fans October 24, / 13
33 P 3 and A P 3 P 3 = (A 4 {0})/G m. So Now So {C P 3 } {(L, s 0,..., s 3 ) s i do not vanish together}. A P 3 = (A 4 {0})/G 4 m. {C A P 3} {((L 0, s 0 ),..., (L 3, s 3 )) s i do not vanish together}. Abramovich (Brown) Artin fans October 24, / 13
34 The monsters evaporate! T P 3 = O 3, Abramovich (Brown) Artin fans October 24, / 13
35 The monsters evaporate! T P 3 = O 3, but T AP 3 = 0. Abramovich (Brown) Artin fans October 24, / 13
36 The monsters evaporate! T P 3 = O 3, but T AP 3 = 0. Logarithmic obstructions to deforming a logarithmic map C A P 3 lie in H 1 (C, 0). Abramovich (Brown) Artin fans October 24, / 13
37 The monsters evaporate! T P 3 = O 3, but T AP 3 = 0. Logarithmic obstructions to deforming a logarithmic map C A P 3 lie in H 1 (C, 0). The obstructions are gone! Abramovich (Brown) Artin fans October 24, / 13
38 Sample theorem Theorem (ℵ-Wise) If Y X is a toric modification, then Logarithmic Gromov Witten invariants of X coincide with those of Y. Abramovich (Brown) Artin fans October 24, / 13
39 Sample theorem Theorem (ℵ-Wise) If Y X is a toric modification, then Logarithmic Gromov Witten invariants of X coincide with those of Y. Reason: M(A Y ) M(A X ) is birational. So M(Y ) M(X ) is virtually birational. Abramovich (Brown) Artin fans October 24, / 13
40 Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ Abramovich (Brown) Artin fans October 24, / 13
41 Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ F P 3 = P 3 F 1 Abramovich (Brown) Artin fans October 24, / 13
42 Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ F P 3 = P 3 F 1 Σ P 3 = TP 3. Abramovich (Brown) Artin fans October 24, / 13
43 Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ F P 3 = P 3 F 1 Σ P 3 = TP 3. X ℶ - Berkovich analytic formal fiber Abramovich (Brown) Artin fans October 24, / 13
44 Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ F P 3 = P 3 F 1 Σ P 3 = TP 3. X ℶ - Berkovich analytic formal fiber P 3 and A P 3 share their tropicalization TP 3. Abramovich (Brown) Artin fans October 24, / 13
45 Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ F P 3 = P 3 F 1 Σ P 3 = TP 3. X ℶ - Berkovich analytic formal fiber P 3 and A P 3 share their tropicalization TP 3. A ℶ X Σ X is a homeomorphism. Abramovich (Brown) Artin fans October 24, / 13
Logarithmic geometry and moduli
Logarithmic geometry and moduli Lectures at the Sophus Lie Center Dan Abramovich Brown University June 16-17, 2014 Abramovich (Brown) Logarithmic geometry and moduli June 16-17, 2014 1 / 1 Heros: Olsson
arxiv: v3 [math.ag] 1 Jun 2017
![arxiv: v3 [math.ag] 1 Jun 2017](/thumbs/80/80532849.jpg "arxiv: v3 [math.ag] 1 Jun 2017")
BIRATIONAL INVARIANCE IN LOGARITHMIC GROMOV WITTEN THEOR DAN ABRAMOVICH AND JONATHAN WISE arxiv:1306.1222v3 [math.ag] 1 Jun 2017 Abstract. Gromov Witten invariants have been constructed to be deformation
Logarithmic geometry and rational curves
Logarithmic geometry and rational curves Summer School 2015 of the IRTG Moduli and Automorphic Forms Siena, Italy Dan Abramovich Brown University August 24-28, 2015 Abramovich (Brown) Logarithmic geometry
LOGARITHMIC STABLE MAPS TO DELIGNE FALTINGS PAIRS I
LOGARITHMIC TABLE MAP TO DELIGNE FALTING PAIR I QILE CHEN Contents 1. Introduction 1 2. Algebricity of the stack of log maps 4 3. Minimal logarithmic maps to rank one Deligne-Faltings log pairs 16 4. The
DECOMPOSITION OF DEGENERATE GROMOV-WITTEN INVARIANTS. Contents 1. Introduction 1
DECOMPOSITION OF DEGENERATE GROMOV-WITTEN INVARIANTS DAN ABRAMOVICH, QILE CHEN, MARK GROSS, AND BERND SIEBERT Abstract. We prove the decomposition formula for stable logarithmic maps. Contents 1. Introduction
Factorization of birational maps for qe schemes in characteristic 0
Factorization of birational maps for qe schemes in characteristic 0 AMS special session on Algebraic Geometry joint work with M. Temkin (Hebrew University) Dan Abramovich Brown University October 24, 2014
WHAT IS TROPICAL GEOMETRY?
WHAT IS TROPICAL GEOMETRY? ERIC KATZ This note was written to answer the question What is tropical geometry? That question can be interpreted in two ways: Would you tell me something about this research
COUNTING CURVES ON TORIC SURFACES TROPICAL GEOMETRY & THE FOCK SPACE
COUNTING CURVES ON TORIC SURFACES TROPICAL GEOMETRY & THE FOCK SPACE RENZO CAVALIERI PAUL JOHNSON HANNAH MARKWIG DHRUV RANGANATHAN ABSTRACT. We study the stationary descendant Gromov Witten theory of toric
Factorization of birational maps on steroids
Factorization of birational maps on steroids IAS, April 14, 2015 Dan Abramovich Brown University April 14, 2015 This is work with Michael Temkin (Jerusalem) Abramovich (Brown) Factorization of birational
arxiv: v2 [math.ag] 22 Dec 2018
![arxiv: v2 [math.ag] 22 Dec 2018](/thumbs/88/117783298.jpg "arxiv: v2 [math.ag] 22 Dec 2018")
NON-ARCHIMEDEAN GEOMETRY OF ARTIN FANS MARTIN ULIRSCH arxiv:1603.07589v2 [math.ag] 22 Dec 2018 Abstract. The purpose of this article is to study the role of Artin fans in tropical and non-archimedean geometry.
Resolving singularities of varieties and families
Resolving singularities of varieties and families Dan Abramovich Brown University Joint work with Michael Temkin and Jaros law W lodarczyk July 2018 Abramovich Resolving singularities of varieties and
LOGARITHMIC MAPS TO DELIGNE-FALTINGS PAIRS
LOGARITHMIC MAP TO DELIGNE-FALTING PAIR DAN ABRAMOVICH AND QILE CHEN Contents 1. Introduction 1 2. Prerequisites on logarithmic geometry 1 3. Logarithmic curves and their stacks 8 4. Algebricity of the
STABLE LOGARITHMIC MAPS TO DELIGNE FALTINGS PAIRS II
STABLE LOGARITHMIC MAPS TO DELIGNE FALTINGS PAIRS II DAN ABRAMOVICH AND QILE CHEN Abstract. We make an observation which enables one to deduce the existence of an algebraic stack of log maps for all generalized
Lifting Tropical Curves and Linear Systems on Graphs
Lifting Tropical Curves and Linear Systems on Graphs Eric Katz (Texas/MSRI) November 7, 2010 Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 1 / 31 Tropicalization Let K = C{{t}} = C((t)),
Logarithmic geometry and moduli
Logarithmic geometry and moduli Dan Abramovich, Qile Chen, Danny Gillam, Yuhao Huang, Martin Olsson, Matthew Satriano, and Shenghao Sun Abstract. We discuss the role played by logarithmic structures in
Mirror symmetry. Mark Gross. July 24, University of Cambridge
University of Cambridge July 24, 2015 : A very brief and biased history. A search for examples of compact Calabi-Yau three-folds by Candelas, Lynker and Schimmrigk (1990) as crepant resolutions of hypersurfaces
Logarithmic Stable Maps with Torus Actions
Logarithmic Stable Maps with Torus Actions by Samouil Molcho B.A., Brandeis University; Waltham, MA, 2009 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of
A tale of Algebra and Geometry
A tale of Algebra and Geometry Dan Abramovich Brown University University of Pisa June 4, 2018 Abramovich (Brown) A tale of Algebra and Geometry June 4, 2018 1 / 12 Intersection theory on algebraic stacks
Infinite root stacks of logarithmic schemes
Infinite root stacks of logarithmic schemes Angelo Vistoli Scuola Normale Superiore, Pisa Joint work with Mattia Talpo, Max Planck Institute Brown University, May 2, 2014 1 Let X be a smooth projective
MA 206 notes: introduction to resolution of singularities
MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k be
THE QUANTUM CONNECTION
THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,
BEZOUT S THEOREM CHRISTIAN KLEVDAL
BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this
Resolving singularities of varieties and families
Resolving singularities of varieties and families Dan Abramovich Brown University Joint work with Michael Temkin and Jaros law W lodarczyk Rio de Janeiro August 7, 2018 Abramovich Resolving varieties and
DIVISOR THEORY ON TROPICAL AND LOG SMOOTH CURVES
DIVISOR THEORY ON TROPICAL AND LOG SMOOTH CURVES MATTIA TALPO Abstract. Tropical geometry is a relatively new branch of algebraic geometry, that aims to prove facts about algebraic varieties by studying
SIMONS SYMPOSIUM 2015: OPEN PROBLEM SESSIONS
SIMONS SYMPOSIUM 2015: OPEN PROBLEM SESSIONS The 2015 Simons Symposium on Tropical and Nonarchimedean Geometry included two open problem sessions, in which leading experts shared questions and conjectures
EXPANDED DEGENERATIONS AND PAIRS DAN ABRAMOVICH, CHARLES CADMAN, BARBARA FANTECHI, AND JONATHAN WISE
EXPANDED DEGENERATIONS AND PAIRS DAN ABRAMOVICH, CHARLES CADMAN, BARBARA FANTECHI, AND JONATHAN WISE Abstract. Since Jun Li s original definition, several other definitions of expanded pairs and expanded
Tropical geometry of logarithmic schemes
Tropical geometry of logarithmic schemes By Martin Ulirsch M.S., Brown University Dipl.-Math., University of Regensburg A dissertation submitted in partial fulfillment of the requirements for the degree
SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS
SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e on a general complete intersection of multidegree
EXCLUDED HOMEOMORPHISM TYPES FOR DUAL COMPLEXES OF SURFACES
EXCLUDED HOMEOMORPHISM TYPES FOR DUAL COMPLEXES OF SURFACES DUSTIN CARTWRIGHT Abstract. We study an obstruction to prescribing the dual complex of a strict semistable degeneration of an algebraic surface.
BRILL-NOETHER THEORY FOR CURVES OF FIXED GONALITY (AFTER JENSEN AND RANGANATHAN)
BRILL-NOETHER THEORY FOR CURVES OF FIXED GONALITY (AFTER JENSEN AND RANGANATHAN) SAM PAYNE Abstract. This talk, presented at the 2017 Simon Symposium on Nonarchimedean and Tropical Geometry, reports on
Overview of classical mirror symmetry
Overview of classical mirror symmetry David Cox (notes by Paul Hacking) 9/8/09 () Physics (2) Quintic 3-fold (3) Math String theory is a N = 2 superconformal field theory (SCFT) which models elementary
arxiv: v1 [math.ag] 30 Mar 2016
![arxiv: v1 [math.ag] 30 Mar 2016](/thumbs/86/94175444.jpg "arxiv: v1 [math.ag] 30 Mar 2016")
arxiv:1603.09213v1 [math.ag] 30 Mar 2016 A GENERALIZATION OF THE DOUBLE RAMIFICATION CYCLE VIA LOG-GEOMETRY JÉRÉMY GUÉRÉ Abstract. We give a log-geometric description of the space of twisted canonical
Collapsing Calabi-Yau Manifolds workshop Talk Schedule
Collapsing Calabi-Yau Manifolds workshop Talk Schedule Events for: Monday, August 31st - Friday, September 4th 10:00am Dave Morrison - SCGP 102 Monday, August 31st Title: The singular fibers in an SYZ
Noncommutative deformations and perverse schobers
Noncommutative deformations and perverse schobers Work in progress with Ludmil Katzarkov Andrew Harder University of Miami January 26, 2017 Andrew Harder (University of Miami) NCD and PSC January 26, 2017
Nested Donaldson-Thomas invariants and the Elliptic genera in String theor
Nested Donaldson-Thomas invariants and the Elliptic genera in String theory String-Math 2012, Hausdorff Center for Mathematics July 18, 2012 Elliptic Genus in String theory: 1 Gaiotto, Strominger, Yin
MATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
Cycle groups for Artin stacks
Cycle groups for Artin stacks arxiv:math/9810166v1 [math.ag] 28 Oct 1998 Contents Andrew Kresch 1 28 October 1998 1 Introduction 2 2 Definition and first properties 3 2.1 The homology functor..........................
arxiv: v4 [math.ag] 29 Jan 2017
![arxiv: v4 [math.ag] 29 Jan 2017](/thumbs/89/98618825.jpg "arxiv: v4 [math.ag] 29 Jan 2017")
FUNCTORIAL TROPICALIZATION OF LOGARITHMIC SCHEMES: THE CASE OF CONSTANT COEFFICIENTS MARTIN ULIRSCH arxiv:1310.6269v4 [math.ag] 29 Jan 2017 Abstract. The purpose of this article is to develop foundational
Cohomology jump loci of quasi-projective varieties
Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)
Oberwolfach Preprints
Oberwolfach Preprints OWP 017-18 RENZO CAVALIERI, PAUL JOHNSON, HANNAH MARKWIG AND DHRUV RANGANATHAN Counting Curves on Toric Surfaces Tropical Geometry & the Fock Space Mathematisches Forschungsinstitut
(1) is an invertible sheaf on X, which is generated by the global sections
7. Linear systems First a word about the base scheme. We would lie to wor in enough generality to cover the general case. On the other hand, it taes some wor to state properly the general results if one
ON TAME STACKS IN POSITIVE CHARACTERISTIC
ON TAME STACKS IN POSITIVE CHARACTERISTIC DAN ABRAMOVICH, MARTIN OLSSON, AND ANGELO VISTOLI Contents 1. Linearly reductive finite group schemes 1 2. Tame stacks 13 3. Twisted stable maps 20 4. Reduction
arxiv:math/ v2 [math.ag] 13 Apr 2008
![arxiv:math/ v2 [math.ag] 13 Apr 2008](/thumbs/93/111380074.jpg "arxiv:math/ v2 [math.ag] 13 Apr 2008")
arxiv:math/0603151v2 [math.ag] 13 Apr 2008 GROMOV WITTEN THEORY OF DELIGNE MUMFORD STACKS DAN ABRAMOVICH, TOM GRABER, AND ANGELO VISTOLI Contents 1. Introduction 1 2. Chow rings, cohomology and homology
RESEARCH STATEMENT: COMPACTIFYING RELATIVE PICARD SCHEMES
RESEARCH STATEMENT: COMPACTIFYING RELATIVE PICARD SCHEMES ATOSHI CHOWDHURY 1. Introduction 1.1. Moduli spaces and compactification. My research is in the area of algebraic geometry concerned with moduli
PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES
PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES Angelo Vistoli Scuola Normale Superiore Bordeaux, June 23, 2010 Joint work with Niels Borne Université de Lille 1 Let X be an algebraic variety over C, x 0 X. What
where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism
8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the
The tangent space to an enumerative problem
The tangent space to an enumerative problem Prakash Belkale Department of Mathematics University of North Carolina at Chapel Hill North Carolina, USA belkale@email.unc.edu ICM, Hyderabad 2010. Enumerative
0.1 Spec of a monoid
These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.
Logarithmic resolution of singularities. Conference in honor of Askold Khovanskii
Logarithmic resolution of singularities M. Temkin July 7, 2017 Conference in honor of Askold Khovanskii M. Temkin (Hebrew University) Logarithmic resolution of singularities 1 / 24 Introduction The ultimate
7.3 Singular Homology Groups
184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the
Deformation theory of representable morphisms of algebraic stacks
Deformation theory of representable morphisms of algebraic stacks Martin C. Olsson School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, molsson@math.ias.edu Received:
Isogeny invariance of the BSD conjecture
Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p
A MORE GENERAL ABC CONJECTURE. Paul Vojta. University of California, Berkeley. 2 December 1998
A MORE GENERAL ABC CONJECTURE Paul Vojta University of California, Berkeley 2 December 1998 In this note we formulate a conjecture generalizing both the abc conjecture of Masser-Oesterlé and the author
Using Stacks to Impose Tangency Conditions on Curves
Using Stacks to Impose Tangency Conditions on Curves arxiv:math/0312349v3 [math.ag] 5 Jul 2005 Charles Cadman Abstract We define a Deligne-Mumford stack X D,r which depends on a scheme X, an effective
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in
SEMI-STABLE DEGENERATIONS AND PERIOD SPACES FOR POLARIZED K3 SURFACES
SEMI-STABLE DEGENERATIONS AND PERIOD SPACES FOR POLARIZED K3 SURFACES MARTIN C. OLSSON Abstract. Modular compactifications of moduli spaces for polarized K3 surfaces are constructed using the tools of
Topology of Nonarchimedean Analytic Spaces
Topology of Nonarchimedean Analytic Spaces AMS Current Events Bulletin Sam Payne January 11, 2013 Complex algebraic geometry Let X C n be an algebraic set, the common solutions of a system of polynomial
MODULI STACKS OF STABLE TORIC QUASIMAPS
MODULI STACKS OF STABLE TORIC QUASIMAPS IONUŢ CIOCAN-FONTANINE AND BUMSIG KIM Abstract. We construct new virtually smooth modular compactifications of spaces of maps from nonsingular curves to smooth projective
Combinatorics and geometry of E 7
Combinatorics and geometry of E 7 Steven Sam University of California, Berkeley September 19, 2012 1/24 Outline Macdonald representations Vinberg representations Root system Weyl group 7 points in P 2
An Atlas For Bun r (X)
An Atlas For Bun r (X) As told by Dennis Gaitsgory to Nir Avni October 28, 2009 1 Bun r (X) Is Not Of Finite Type The goal of this lecture is to find a smooth atlas locally of finite type for the stack
Gromov-Witten invariants and Algebraic Geometry (II) Jun Li
Gromov-Witten invariants and Algebraic Geometry (II) Shanghai Center for Mathematical Sciences and Stanford University GW invariants of quintic Calabi-Yau threefolds Quintic Calabi-Yau threefolds: X =
SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS
SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree
Hodge Theory of Maps
Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar
ORBIFOLDS AND ORBIFOLD COHOMOLOGY
ORBIFOLDS AND ORBIFOLD COHOMOLOGY EMILY CLADER WEDNESDAY LECTURE SERIES, ETH ZÜRICH, OCTOBER 2014 1. What is an orbifold? Roughly speaking, an orbifold is a topological space that is locally homeomorphic
Parabolic sheaves on logarithmic schemes
Available online at www.sciencedirect.com Advances in Mathematics 231 (2012) 1327 1363 www.elsevier.com/locate/aim Parabolic sheaves on logarithmic schemes Niels Borne a,, Angelo Vistoli b a Laboratoire
LOG SMOOTHNESS AND POLYSTABILITY OVER VALUATION RINGS
LOG SMOOTHNESS AND POLYSTABILITY OVER VALUATION RINGS KARIM ADIPRASITO, GAKU LIU, IGOR PAK, AND MICHAEL TEMKIN Abstract. Let O be a valuation ring of height one of residual characteristic exponent p and
Parabolic sheaves, root stacks and the Kato-Nakayama space
Parabolic sheaves, root stacks and the Kato-Nakayama space Mattia Talpo UBC Vancouver February 2016 Outline Parabolic sheaves as sheaves on stacks of roots, and log geometry. Partly joint with A. Vistoli,
arxiv: v2 [math.co] 25 Dec 2014
![arxiv: v2 [math.co] 25 Dec 2014](/thumbs/90/101988830.jpg "arxiv: v2 [math.co] 25 Dec 2014")
THE NUMBER OF VERTICES OF A TROPICAL CURVE IS BOUNDED BY ITS AREA TONY YUE YU arxiv:1306.3497v2 [math.co] 25 Dec 2014 Abstract. We introduce the notion of tropical area of a tropical curve defined in an
POINCARÉ INVARIANTS ARE SEIBERG-WITTEN INVARIANTS
POINCARÉ INVARIANS ARE SEIBERG-WIEN INVARIANS HUAI-LIANG CHANG AND YOUNG-HOON KIEM Abstract. We prove a conjecture of Dürr, Kabanov and Okonek which provides an algebro-geometric theory of Seiberg-Witten
The moduli stack of vector bundles on a curve
The moduli stack of vector bundles on a curve Norbert Hoffmann norbert.hoffmann@fu-berlin.de Abstract This expository text tries to explain briefly and not too technically the notions of stack and algebraic
UNIVERSAL LOG STRUCTURES ON SEMI-STABLE VARIETIES
UNIVERSAL LOG STRUCTURES ON SEMI-STABLE VARIETIES MARTIN C. OLSSON Abstract. Fix a morphism of schemes f : X S which is flat, proper, and fiber-by-fiber semi-stable. Let IV LS be the functor on the category
ALGEBRAIC COBORDISM REVISITED
ALGEBRAIC COBORDISM REVISITED M. LEVINE AND R. PANDHARIPANDE Abstract. We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main
Generating functions and enumerative geometry
Generating functions and enumerative geometry Ragni Piene Centre of Mathematics for Applications and Department of Mathematics, University of Oslo October 26, 2005 KTH, Stockholm Generating functions Enumerative
arxiv: v2 [math.ag] 29 Aug 2009
![arxiv: v2 [math.ag] 29 Aug 2009](/thumbs/88/117902814.jpg "arxiv: v2 [math.ag] 29 Aug 2009")
LOGARITHMIC GEOMETRY, MINIMAL FREE RESOLUTIONS AND TORIC ALGEBRAIC STACKS arxiv:0707.2568v2 [math.ag] 29 Aug 2009 ISAMU IWANARI Abstract. In this paper we will introduce a certain type of morphisms of
Mini-Course on Moduli Spaces
Mini-Course on Moduli Spaces Emily Clader June 2011 1 What is a Moduli Space? 1.1 What should a moduli space do? Suppose that we want to classify some kind of object, for example: Curves of genus g, One-dimensional
Geometric Class Field Theory
Geometric Class Field Theory Notes by Tony Feng for a talk by Bhargav Bhatt April 4, 2016 In the first half we will explain the unramified picture from the geometric point of view, and in the second half
THE CANONICAL PENCILS ON HORIKAWA SURFACES
THE CANONICAL PENCILS ON HORIKAWA SURFACES DENIS AUROUX Abstract. We calculate the monodromies of the canonical Lefschetz pencils on a pair of homeomorphic Horikawa surfaces. We show in particular that
TROPICAL COMPLEXES DUSTIN CARTWRIGHT
TROPICAL COMPLEXES DUSTIN CARTWRIGHT Abstract. We introduce tropical complexes, which are -complexes together with additional numerical data. On a tropical complex, we define divisors and linear equivalence
Gromov-Witten theory of A n -resolutions
Gromov-Witten theory of A n -resolutions arxiv:82.2681v1 [math.ag] 19 Feb 28 Davesh Maulik February 16, 213 Abstract We give a complete solution for the reduced Gromov-Witten theory of resolved surface
JOHAN S PROBLEM SEMINAR
JOHAN S PROBLEM SEMINAR SHIZHANG LI I m writing this as a practice of my latex and my english (maybe also my math I guess?). So please tell me if you see any mistake of these kinds... 1. Remy s question
DIVISOR CLASSES AND THE VIRTUAL CANONICAL BUNDLE FOR GENUS 0 MAPS
DIVISOR CLASSES AND THE VIRTUAL CANONICAL BUNDLE FOR GENUS 0 MAPS A. J. DE JONG AND JASON STARR Abstract. We prove divisor class relations for families of genus 0 curves and used them to compute the divisor
THE LANDAU-GINZBURG/CALABI-YAU CORRESPONDENCE
THE LANDAU-GINZBURG/CALABI-YAU CORRESPONDENCE EMILY CLADER WEDNESDAY LECTURE SERIES, ETH ZÜRICH, OCTOBER 2014 Given a nondegenerate quasihomogeneous polynomial W (x 1,..., x N ) with weights c 1,..., c
The transverse index problem for Riemannian foliations
The transverse index problem for Riemannian foliations John Lott UC-Berkeley http://math.berkeley.edu/ lott May 27, 2013 The transverse index problem for Riemannian foliations Introduction Riemannian foliations
MATH 221 NOTES BRENT HO. Date: January 3, 2009.
MATH 22 NOTES BRENT HO Date: January 3, 2009. 0 Table of Contents. Localizations......................................................................... 2 2. Zariski Topology......................................................................
The moduli space of curves is rigid
The moduli space of curves is rigid Paul Hacking August 6, 2008 Abstract We prove that the moduli stack M g,n of stable curves of genus g with n marked points is rigid, that is, has no infinitesimal deformations.
mult V f, where the sum ranges over prime divisor V X. We say that two divisors D 1 and D 2 are linearly equivalent, denoted by sending
2. The canonical divisor In this section we will introduce one of the most important invariants in the birational classification of varieties. Definition 2.1. Let X be a normal quasi-projective variety
ON (LOG) TWISTED CURVES. 1. Introduction
ON (LOG) TWISTED CURVES MARTIN C. OLSSON Abstract. We describe an equivalence between the notion of balanced twisted curve introduced by Abramovich and Vistoli, and a new notion of log twisted curve, which
COUNTING ELLIPTIC PLANE CURVES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 12, December 1997, Pages 3471 3479 S 0002-9939(97)04136-1 COUNTING ELLIPTIC PLANE CURVES WITH FIXED j-invariant RAHUL PANDHARIPANDE (Communicated
PICARD GROUPS OF MODULI PROBLEMS II
PICARD GROUPS OF MODULI PROBLEMS II DANIEL LI 1. Recap Let s briefly recall what we did last time. I discussed the stack BG m, as classifying line bundles by analyzing the sense in which line bundles may
Artin-Schelter regular algebras and the Steenrod algebra
Artin-Schelter regular algebras and the Steenrod algebra J. H. Palmieri and J. J. Zhang University of Washington Los Angeles, 10 October 2010 Exercise Let A(1) be the sub-hopf algebra of the mod 2 Steenrod
IV. Birational hyperkähler manifolds
Université de Nice March 28, 2008 Atiyah s example Atiyah s example f : X D family of K3 surfaces, smooth over D ; X smooth, X 0 has one node s. Atiyah s example f : X D family of K3 surfaces, smooth over
MODULI SPACES AND DEFORMATION THEORY, CLASS 1. Contents 1. Preliminaries 1 2. Motivation for moduli spaces Deeper into that example 4
MODULI SPACES AND DEFORMATION THEORY, CLASS 1 RAVI VAKIL Contents 1. Preliminaries 1 2. Motivation for moduli spaces 3 2.1. Deeper into that example 4 1. Preliminaries On the off chance that any of you
ALGEBRAIC COBORDISM REVISITED
ALGEBRAIC COBORDISM REVISITED M. LEVINE AND R. PANDHARIPANDE Abstract. We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main
LOGARITHMIC GEOMETRY AND ALGEBRAIC STACKS. 1. Introduction
LOGARITHMIC GEOMETRY AND ALGEBRAIC STACKS MARTIN C. OLSSON Abstract. We construct algebraic moduli stacks of log structures and give stack-theoretic interpretations of K. Kato s notions of log flat, log
A CRITERION FOR A DEGREE-ONE HOLOMORPHIC MAP TO BE A BIHOLOMORPHISM. 1. Introduction
A CRITERION FOR A DEGREE-ONE HOLOMORPHIC MAP TO BE A BIHOLOMORPHISM GAUTAM BHARALI, INDRANIL BISWAS, AND GEORG SCHUMACHER Abstract. Let X and Y be compact connected complex manifolds of the same dimension
Donaldson-Thomas invariants
Donaldson-Thomas invariants Maxim Kontsevich IHES Bures-sur-Yvette, France Mathematische Arbeitstagung June 22-28, 2007 Max-Planck-Institut für Mathematik, Bonn, Germany 1 Stability conditions Here I ll
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
arxiv:math/ v1 [math.ag] 18 Oct 2003
![arxiv:math/ v1 [math.ag] 18 Oct 2003](/thumbs/93/111435271.jpg "arxiv:math/ v1 [math.ag] 18 Oct 2003")
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp. 139 152. Printed in India The Jacobian of a nonorientable Klein surface arxiv:math/0310288v1 [math.ag] 18 Oct 2003 PABLO ARÉS-GASTESI
arxiv: v3 [math.ag] 31 Jan 2016
![arxiv: v3 [math.ag] 31 Jan 2016](/thumbs/96/128064357.jpg "arxiv: v3 [math.ag] 31 Jan 2016")
EQUATIONS OF TROPICAL VARIETIES JEFFREY GIANSIRACUSA 1 AND NOAH GIANSIRACUSA 2 arxiv:1308.0042v3 [math.ag] 31 Jan 2016 ABSTRACT. We introduce a scheme-theoretic enrichment of the principal objects of tropical
Math 248B. Applications of base change for coherent cohomology
Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).