Rationality of Varieties

Contents
Rationality of Algebraic Varieties
On Geometry of Fano Threefold Hypersurfaces
	1. Introduction
	2. Non-solid Fano threefolds
	3. Birationally non-rigid Fano threefolds
		3.1. How to read off the equation of Z?
	4. Evidence for Conjecture 1.4
		Acknowledgement
	References
On the Image of the Second l-adic Bloch Map
	Introduction
		0.1. Mazur\'s question with Q-coefficients
		0.2. Mazur\'s question with Q-coefficients in positive characteristic
		0.3. Mazur\'s question with Z-coefficients
		0.4. Universal cycles and the image of the second l-adic Bloch map
		0.5. Decomposition of the diagonal and the image of the second l-adic Bloch map
		0.6. Stably rational vs. geometrically stably rational varieties over finite fields
		0.7. Notation and conventions
	1. On various notions of coniveau filtrations
		1.1. Recalling the geometric coniveau filtrations
		1.2. p-adic coniveau filtrations
	2. The image of the l-adic Bloch map and the coniveau filtration
		2.1. The image of the -adic Bloch map
		2.2. The image of the p-adic Bloch map
	3. Decomposition of the diagonal, algebraic representatives,and miniversal cycles
		3.1. Decomposition of the diagonal
		3.2. Surjective regular homomorphisms and algebraic representatives
		3.3. Miniversal cycles and miniversal cycles of minimal degree
		3.4. Decomposition of the diagonal and algebraic representatives
	4. Miniversal cycles and the image of the second l-adic Bloch map
	5. Decomposition of the diagonal and the image of the second -adic Bloch map
	6. Modeling cohomology via correspondences
		6.1. Modeling Q-cohomology via correspondences
		6.2. Modeling Z-cohomology via correspondences: Theorem 15
	7. The image of the -adic Bloch map in characteristic 0
	Appendix: A review of the l-adic Bloch map
		A.1. Conventions for -adic and p-adic cohomology
			A.1.1. -adic cohomology
			A.1.2. p-adic cohomology
		A.2. The -adic Bloch map
			A.2.1. The Abel–Jacobi map on torsion
			A.2.2. Bloch\'s preliminaries
			A.2.3. The -Bloch map
			A.2.4. Bloch\'s Key Lemma
			A.2.5. The -adic Bloch map
			A.2.6. The Bloch map
		A.3. Suwa\'s construction of the l-adic Bloch map
			A.3.1. Structure of abelian l-primary torsion groups
			A.3.2. -adic cohomology from cohomology with torsion coefficients
			A.3.3. Suwa\'s -adic Bloch map
			A.3.4. The -adic Bloch map and Suwa\'s construction
			A.3.5. Gross–Suwa\'s p-adic Bloch map
		A.4. Properties of the Bloch maps
		A.5. Restriction of the Bloch map to algebraically trivial cycle classes
		Acknowledgment
	References
Rational Curves and MBM Classes on Hyperkähler Manifolds: A Survey
	1. Introduction
	2. MBM classes: equivalent definitions and basic properties
		2.1. Deforming rational curves: first remarks
		2.2. Parameter spaces for hyperkähler manifolds
		2.3. MBM classes
	3. Results on MBM classes and applications
		3.1. Markman\'s Torelli theorem and the birational cone conjecture
		3.2. The cone conjecture via ergodic theory
		3.3. Uniform boundedness and an appication
	4. Contractibility and deformations
	5. Classification of MBM classes in low dimension for K3 type
	6. Some open questions
		Acknowledgement
	References
Unirationality of Certain Universal Families of Cubic Fourfolds
	1. Introduction
	2. The existence of the universal cubic fourfold, and some properties of scrolls and associated K3 surfaces
	3. Unirationality for C26,1 and C42,1 via universal K3 surfaces
	4. Unirationality through rational special surfaces
		4.1. Special cubics in Cd in the range 8d 38
		4.2. Special cubics in C42
		4.3. Unirationality of Cd,n
	5. Some results of non-unirationality
		5.1. Open questions
		Acknowledgement
	References
A Categorical Invariant for Geometrically Rational Surfaces with a Conic Bundle Structure
	1. Introduction
		Notations
	2. Basics on geometrically rational surfaces
		2.1. Elementary links
	3. Basics on derived categories
		3.1. Categorical representability
		3.2. Conic bundles
	4. Links of type I/III and the definition ofthe Griffiths–Kuznetsov component
	5. Links of type II
	6. Links of type IV
		Acknowledgment
	References
Marked and Labelled Gushel–Mukai Fourfolds
	1. Introduction
	2. Gushel–Mukai fourfolds
		2.1. Cohomology and period domain of Gushel–Mukai fourfolds
		2.2. Hodge-special Gushel–Mukai fourfolds
	3. Marked and labelled Gushel–Mukai fourfolds
	4. Gushel–Mukai fourfolds with associated K3 surface
		4.1. Rational maps to moduli spaces of K3 surfaces
		4.2. Fibers of Fourier–Mukai partners
	5. Gushel–Mukai fourfolds and twisted K3 surfaces
		5.1. Moduli and periods of twisted K3 surfaces
		5.2. Twisted K3 surfaces associated to GM fourfolds
		5.3. Fourier-Mukai partners in the twisted case
		Acknowledgment
	References
Supersingular Irreducible Symplectic Varieties
	1. Introduction
	2. Generalities on the notion of supersingularity
	3. Supersingular symplectic varieties
	4. Moduli spaces of stable sheaves on K3 surfaces
	5. Moduli spaces of twisted sheaves on K3 surfaces
	6. Moduli spaces of sheaves on abelian surfaces
	References
Symbols and Equivariant Birational Geometry in Small Dimensions
	1. Brief history of previous work
	2. Equivariant birational types
		2.1. Antisymmetry
		2.2. Multiplication and co-multiplication
		2.3. Birational invariant
	3. Computation of invariants on surfaces
		3.1. Sample computations of B2(Cp)
		3.2. Examples for noncyclic groups
		3.3. Linear actions yield torsion classes
		3.4. Algebraic structure in dimension 2
	4. Reconstruction theorem
	5. Refined invariants
		5.1. Encoding fixed points
		5.2. Encoding points with nontrivial stabilizer
		5.3. Examples of blowup relations
		5.4. Examples
		5.5. Limitation of the birational invariant
		5.6. Reprise: Cyclic groups on rational surfaces
	6. Cubic fourfolds
	7. Nonabelian invariants
		7.1. The equivariant Burnside group
		7.2. Resolution of singularities
		7.3. The class of XG
		7.4. Elementary observations
		7.5. Dihedral group of order 12
		7.6. Embeddings of S3C2 into the Cremona group
		Acknowledgment
	References
Rationality of Fano Threefolds of Degree 18 over Non-closed Fields
	1. Introduction
	2. Projection constructions
		2.1. Projection from lines
		2.2. Projection from conics
		2.3. Projection from points
	3. Unirationality constructions
		3.1. Using a point
		3.2. Using a point and a conic
	4. Rationality results
	5. Analysis of principal homogeneous spaces
		5.1. Proof of Theorem 1
		5.2. A corollary to Theorem 1
		5.3. Generic behavior
		5.4. Connections with complete intersections?
		Acknowledgment
	References
Rationality of Mukai Varieties over Non-closed Fields
	1. Introduction
	2. A birational transformation given by a family of quadrics
		2.1. The statement
		2.2. The proof
		2.3. Grassmannians of lines
		2.4. Orthogonal Grassmannian
		2.5. Grassmannian of the group G2
	3. Mukai varieties of genus 7, 8, and 10
		3.1. Forms of linear sections
		3.2. Rationality of Mukai varieties
	4. Mukai varieties of genus 9
		4.1. The statement
		4.2. The proof
		4.3. Implications for genus 9 Mukai varieties
	5. Fano threefolds of genus 12
		5.1. Vector bundles and Grassmannian embedding
		5.2. Birational transformation for `39`42`\"613A``45`47`\"603AGr(3,7)
		5.3. The induced transformation of threefolds
	Appendix: Application to cylinders
		Acknowledgment
	References
A Refinement of the Motivic Volume, and Specialization of Birational Types
	1. Introduction
		Terminology
	2. The Grothendieck ring of varieties graded by dimension
		2.1. Reminders on the Grothendieck ring of varieties
		2.2. The graded Grothendieck ring
		2.3. Birational types
		2.4. A refinement of Bittner\'s presentation
		2.5. A refinement of the theorem of Larsen & Lunts
	3. Dimensional refinement of the motivic volume
		3.1. The motivic volume
		3.2. Strictly toroidal models
		3.3. Construction of the motivic volume
	4. Applications to rationality problems
		4.1. Specialization of birational types
		4.2. Obstruction to stable rationality
		4.3. Examples
	5. The monodromy action
		5.1. The equivariant Grothendieck ring
		5.2. The monodromy action on the motivic volume
		Acknowledgement
	References
Explicit Rationality of Some Special Fano Fourfolds
	Introduction
	1. Rationality via linear systems of hypersurfaces of degree 3e-1 having points of multiplicity e along a surface
		1.1. Linear systems of quintics with double points along a general Sd
	2. Birational maps to P4 for cubics in C14, C26 and C38
	3. Birational maps to linear sections of G(1,3+k) for cubics in C(14+12k) for k <= 2
	4. A divisor of rational Gushel–Mukai fourfolds
		4.1. Del Pezzo fivefolds through a K3 surface of degree 14 and genus 8
		4.2. GM fourfolds through a K3 surface of degree 14 and genus 8
		4.3. Surfaces of degree 10 and sectional genus 6 with a node in P5 obtained as projections of general K3 surfaces of degree 10 and genus 6
		4.4. Rationality of the GM fourfolds in p-1(D10\')
	5. Computations via Macaulay2
		Acknowledgement
	References
Unramified Cohomology, Algebraic Cycles and Rationality
	1. Introduction
		1.1. Notation and convention
	2. Preliminaries from étale cohomology
		2.1. Cohomology of fields
		2.2. Commutativity with direct limits
		2.3. Long exact sequence of pairs
		2.4. Cup products
		2.5. Gysin sequence
	3. Residue maps
		3.1. Compatibility with pullbacks
		3.2. Compatibility with pushforwards
		3.3. A consequence of Bloch–Ogus\' theorem
	4. Unramified cohomology
		4.1. Stable invariance
		4.2. Functoriality
		4.3. Restriction to scheme points and pullbacks for morphisms betweensmooth projective varieties
		4.4. It is enough to check residues on a smooth proper model
		4.5. Comparison with usual cohomology
	5. Merkurjev\'s pairing
		5.1. Applications of Proposition 5.1
		5.2. Proof of Proposition 5.1
	6. Generalization to schemes with normal crossings
		6.1. A pairing on the level of 0-cycles
		6.2. A pairing on the level of correspondences
	7. Decompositions of the diagonal
		7.1. Connection to rationality and stable birational types
		7.2. Torsion orders
	8. Specialization method
	9. Examples with nontrivial unramified cohomology
		9.1. Quadric bundles á la Artin–Mumford and Colliot-Thélène–Ojanguren
		9.2. The quadric surface bundle of Hassett–Pirutka–Tschinkel
		9.3. Generalization
	10. Vanishing result and applications
	11. Open problems
		11.1. Decompositions of the diagonal versus stable rationality
		11.2. Torsion orders and unirationality
		Acknowledgement
	References
Vanishing Cycles under Base Change and the Integral Hodge Conjecture
	1. Introduction
	2. Vanishing cycles under blow-up
		2.1. An induction process
		2.2. Applications to vanishing cycles
			2.2.1. Local situation
			2.2.2. Global situation
	3. Applications to the integral Hodge conjecture
		3.1. Special case: hypersurfaces
		3.2. General case: hyperplane sections
		Acknowledgment
	References
The Igusa Quartic and the Prym Map, with Some Rational Moduli
	1. Introduction and preliminaries
	2. fffPlane geometry in P4: the congruence S and its focal locus B
	3. Conic bundles associated to a genus 6 Prym curve
	4. Igusa pencils and E6-quartic threefolds
	5. Moduli of Prym sextics of genus 6
	6. Moduli of Igusa pencils and of E6-quartics
	7. Revisiting the Prym map
	8. The period map j and the universal set of double sixes
		Acknowledgment
	References