The Art of Doing Algebraic Geometry

Preface
Contents
Weyl Cycles on the Blow-Up of double struck upper P Superscript 4mathbbP4 at Eight Points
1 Introduction
2 Preliminaries
3 Weyl Cycles in double struck upper P Superscript nmathbbPn Blown Up at ss Points
3.1 Weyl Curves
4 double struck upper P cubedmathbbP3 Blown Up in Seven Points
5 double struck upper P Superscript 4mathbbP4 Blown Up in Eight Points
5.1 Curves in upper X 8 Superscript 4X48
5.2 Further Blow-up of double struck upper P Superscript 4mathbbP4
5.3 Classification of the Weyl Surfaces
5.4 Weyl Divisors
6 Weyl Expected Dimension
References
Simson’s Reconstruction of Apollonius’ Loci Plani. Modern Ideas in Classical Language
1 Introduction
2 Harmonic Group, Tangent and Polar in Apollonius’ Conics
3 Book I of Loci Plani
3.1 A, B, C Aligned, a Given Ratio Between AB and AC (Central Similitude)
3.2 A, B, C Aligned; AB and AC Inversely Proportional; B on a Given Line
3.3 A, B, C Aligned; AB and AC Inversely Proportional; B on a Given Circumference
3.4 AB and AC Forming a Given Angle and Having a Given Ratio; B on a Given Line or on a Given Circumference. (Proposition 6)
4 Book II of Loci Plani
5 Simson’s Preface
6 Conclusions
References
Kummer Quartic Surfaces, Strict Self-duality, and More
1 Introduction
1.1 Notation
2 Proof of Theorem 3 Over double struck upper CmathbbC via Theta Functions
3 Segre's Construction and Special Kummer Quartics
3.1 Proof of Theorem 5
4 Proof of Theorem 1
5 Algebraic Proofs for Theorems 4 and 3
6 The Role of the Segre Cubic Hypersurface
7 Enriques Surfaces Étale Quotients of Kummer K3 Surfaces
8 Remarks on Normal Cubic Surfaces and on Strict Selfduality
9 Appendix on Monoids
References
A Footnote to a Footnote to a Paper of B. Segre
1 Introduction
2 Preliminaries
2.1 Notation
2.2 Catalecticant Maps
2.3 Hilbert Functions
2.4 Ramification and the Terracini Locus
3 Plane Sextics
4 Equations for Loci in upper S Superscript 9S9
References
Deformations and Extensions of Gorenstein Weighted Projective Spaces
1 Introduction
2 Gorenstein Weighted Projective Spaces
3 Extendability of Non-primitive Polarized upper K Baseline 3K3 Surfaces
4 Extendability and Graded Deformations of Cones
5 The Deformation Argument
6 Explicit Computations on WPS
7 Examples
References
Intersection Cohomology and Severi Varieties
1 Introduction
2 Notations and Basic Facts
3 Local Study of the Dual Hypersurface
4 Intersection Cohomology Complex on Severi Varieties
5 Examples
5.1 Curves in a Projective Surface
5.2 Defective Hypersurfaces of double struck upper P Superscript 2 nmathbbP 2n
5.3 Defective Hypersurfaces in a Complete Intersection of Quadrics
References
Cremona Orbits in double struck upper P Superscript 4mathbbP4 and Applications
1 Introduction
2 The Standard Cremona Transformation and Its Resolution
3 The Case of Three Space
4 The Chow Ring for the Case of double struck upper P Superscript 4mathbbP4
5 The Cremona Involution on double struck upper P Superscript 4mathbbP4
6 Six and Seven Points in double struck upper P Superscript 4mathbbP4
7 Eight Points in double struck upper P Superscript 4mathbbP4
8 Applications
References
On Some Components of Hilbert Schemes of Curves
1 Generalities
1.1 Hilbert Schemes and Brill–Noether Theory of Curves
1.2 Gaussian–Wahl Maps and Cones
1.3 Ramified Coverings of Curves
2 Curves and Cones
2.1 Curves in Principal Components
2.2 Cones Extending Curves in ModifyingAbove script upper I With caretmathcalI"0362mathcalI
2.3 Curves on Cones and Ramified Coverings
3 Superabundant Components of Hilbert Schemes
References
Siegel Modular Forms of Degree Two and Three and Invariant Theory
1 Introduction
2 Siegel Modular Forms
3 Moduli of Curves of Genus Two as a Stack Quotient
4 Invariant Theory of Binary Sextics
5 Covariants of Binary Sextics and Modular Forms
6 Constructing Vector-Valued Modular Forms of Degree 22
7 Rings of Scalar-Valued Modular Forms
8 Moduli of Curves of Genus Three and Invariant Theory of Ternary Quartics
9 Concomitants of Ternary Quartics and Modular Forms of Degree 33
References
On Intrinsic Negative Curves
1 Introduction
2 Intrinsic Curves
3 Infinite Families
4 Seshadri Constants
4.1 Weighted Projective Planes
References
On the Extendability of Projective Varieties: A Survey
1 Introduction
2 Extendability in General
3 How to Estimate alpha left parenthesis upper X right parenthesisα(X): A Fortunate Coincidence
4 Old Days
5 Extendability of Canonical Curves, K3 Surfaces and Fano Threefolds
6 Extendability of Enriques Surfaces and Enriques-Fano Threefolds
7 Extendability of Curves in Other Embeddings
8 Extendability of Other Surfaces
8.1 Surfaces of Kodaira Dimension negative normal infinity-infty
8.2 Surfaces of Kodaira Dimension 00
8.3 Surfaces of Kodaira Dimension 11
8.4 Surfaces of General Type
9 Appendix: Extendability of Canonical Models of Plane Quintics
References
The Minimal Cremona Degree of Quartic Surfaces
1 Introduction
2 History and Background
3 Minimal Cremona Degree of Quartics
References
On the Degree of the Canonical Map of a Surface of General Type
1 Introduction
2 The Canonical Map
2.1 The Canonical Image
2.2 Bounds on the Canonical Degree
3 Two Constructions
3.1 Generating Pairs
3.2 Abelian Covers
4 Examples
4.1 Examples Arising from Generating Pairs
4.2 Examples Constructed as Abelian Covers
5 Remarks and Open Questions
5.1 Case (A) of Theorem 2.1 (p Subscript g Baseline left parenthesis normal upper Sigma right parenthesis equals 0pg(Σ)=0)
5.2 Case (B) of Theorem 2.1 (p Subscript g Baseline left parenthesis normal upper Sigma right parenthesis equals p Subscript g Baseline left parenthesis upper X right parenthesispg(Σ)=pg(X))
References
Hyper-Kähler Varieties with a Motive of Abelian Type
1 Introduction
2 The Motive of the HK Variety upper ZZ Constructed in ch14LLSS
3 The Motive of the Compactification script upper J overbarbarmathcalJ in ch14LSV
References
Finite Quotients of Surface Braid Groups and Double Kodaira Fibrations
1 Introduction
2 Pure Surface Braid Groups and Finite Braid Quotients
3 Extra-Special Groups as Pure Braid Quotients
4 Geometrical Application: Diagonal Double Kodaira Fibrations
5 Beyond |G|=32
References
Affine Cones over Fano–Mukai Fourfolds of Genus 1010 are Flexible
1 Introduction
1.1 Flexible Varieties
1.2 Main Results
2 Cubic Scrolls in the Fano–Mukai Fourfolds upper V 18V18
3 Criteria of Flexibility of Affine Cones
4 normal upper A normal u normal t Superscript 0 Baseline left parenthesis upper V right parenthesisAut0(V)-Action on the Fano–Mukai Fourfold upper VV of Genus 10
5 Affine 44-Spaces in upper V 18 Superscript normal a Va18 and Flexibility of Affine Cones
6 double struck upper A squaredmathbbA2-Cylinders in Smooth Quadric Fourfolds and in the Del Pezzo Quintic Fourfold
7 double struck upper A squaredmathbbA2-Cylinders in upper V 18V18 and Flexibility of Affine Cones over upper V 18 Superscript normal aVa18
References
Enriques Diagrams Under Pullback by a Double Cover
1 Clusters and Enriques Diagrams
1.1 The Algorithm
1.2 Infintely Near Points Verses Divisorial Valuations
2 Double Covers
2.1 Effect on Valuations
2.2 The Main Trunk and Its Pullback
2.3 Classification of Infinitely Near Points on upper S primeS'
3 Justification for the Algorithm
3.1 Additional Remarks
References
The ``Projective Spirit'' in Segre's Lectures on Differential Equations
1 The ``Projective Spirit'' of Italian Mathematics, 1860–1940
2 Segre's Notebooks
3 The Projective Point of View in the Theory of Differential Equations
4 A Taste of Segre's Lectures on Differential Equations
4.1 Jacobi Equations and upper WW-curves
4.2 Congruences of Curves and Systems of ODE
5 Conclusions
References