05-20-2022

Let \(X\) be a K3 surface.

Basics

What is \(\chi(X)\)?
What is \(c_1(X)\)? \(c_2(X)\)?
What is the lattice decomposition of \(H^2(X; {\mathbb{Z}})\)? What is its signature?
What is the Hodge diamond for \(X\)?
What is the normalization?
What is a special fiber?
What is a central fiber?
What are normal crossings?
What is an \(n{\hbox{-}}\)curve?
What does it mean to contract a curve?
What is the Torelli theorem?
What is a double curve?
What is a triple point?
What is properness?
What is semistability?
What are Weil divisors?
What is \({\mathcal{O}}(n), {\mathcal{O}}(-n)\)?
What are extended Dynkin diagrams?
What is the degree of a K3?

Reading

What is a minimal resolution?
What is a Kulikov surface?
What is a Looijenga semitoroidal compactification?
What is a polarized K3?
What is \(F_{2d}\)?
- What is the Bailey-Borel compactification?
- What is an admissible fan?
- What are the associated toroidal compactifications?
- What is a toroidal compactification?
What is a theta divisor?
What is an slc singularity?
What is a stable pair?
What is an ample divisor?
What is the linear system associated to a divisor?
What is the rational curve divisor?
What is the Yau-Zaslow formula?
Why is a generic K3 a double branched cover of \({\mathbb{P}}^2\)?
What is the flex divisor?
What is the Kummer variety of a PPAV?
What are some examples of K3s?
What is the monodromy invariant?
What is a polarization divisor?
What is a big and nef class \(L\in M \subset {\operatorname{Pic}}(X)\)?
What is a unimodular lattice?
- What is the signature of a lattice?
What is the Lefschetz (1, 1) theorem?
What is Neron-Severi?
Why does \({\operatorname{NS}}(X) = {\operatorname{Pic}}(X)\) here?
What are the roots of \(X\)?
What is the Weyl group of \(X\)?
Why does every K3 admit a Kahler form?
What is the Kahler cone?
What are the lattices \(H\) and \(E_8\)?
What is a marking for a smooth family of K3s?
What is the Gauss-Manin connection?
- How is its monodromy computed?
What is the period domain?
What is the period map?
What is a Kuranashi family?
- Why does one exist for \(X\)?
What is a universal deformation?
How is the moduli of K3s built by gluing deformation spaces?
What is the coarse moduli space for K3s? The fine moduli space?
What is a lattice-polarized K3?
What is a Weyl chamber?
What is an M-quasipolarized K3 surface?
What is an M-quasipolarized period domain?
What is \({\mathbb{D}}_M\)?
What is the Weyl group \(W_x\) for \(x\in {\mathbb{D}}_M\)?
What are the small cones?
What does it mean for a lattice to be hyperbolic?
What does it mean to be locally rationally polyhedral?
What is a Type IV Hermitian symmetric domain?
What is a positive root in \({\operatorname{NS}}(X)\)? A simple root?
- Why do positive roots represent effective divisors? What do the simple roots represent?
What is a rational double point of ADE type?
What is the inertia stack?
What is a complex orbifold?
How is a quotient stack defined?
What is a smooth stack?
What is the analytic germ of a smooth curve?
What are Kulikov models of types I, II, III?
What is a Kulikov surface?
What is the Enriques-Kodaira classification?
What is a smooth elliptic anticanonical double curve?
What is a rational surface?
What is an anticanonical pair?
What is a reduced nodal anticanonical divisor?
What is the strict transform?
What is the dual complex?
What is an example of a Type II Kulikov surface?
How do you glue surfaces along fibers?
What is a reduced normal crossings surface?
What is d-semistability?
What does it mean to be smoothable?
What is the charge of an anticanonical pair?
What are corner and internal blowups?
What is a toric pair?
What is a toric boundary?
What is a toric model?
What is an order \(k\) branched cover?
What is a ruled surface?
What are the local coordinates near a double curve and a triple point?
What is a Kulikov degeneration?
What is a quasipolarization on the general fiber?
What does it mean to be relatively nef?
What is a nef model? Why does it exist?
What is a divisor model? Why does it exist?
What is a stable model?
What is the theory of canonical models?
What does it mean to be relatively ample?
What are log canonical singularities?
What are semi log canonical singularities?
What is a resolution of singularities?
What does it mean to have ADE singularities?
What is the Picard-Lefschetz transformation?
What does it mean to be unipotent?
What is a primitive isotropic vector?
Why is the log monodromy integral?
What is imprimitivity?
How are the types of Kulikov models distinguished?
What is a simple normal crossings degeneration?
What is the Clemens collapse?
What is the arithmetic genus? The geometric genus?
What is a Cartier divisor?
What is a mixed Hodge structure on a lattice?
What is the cocharacter lattice?
What is an Atiyah flop along a curve?
What is the Noether-Lefschetz locus?
What is Zariski’s main theorem?
What is a stable pair?
What are slc singularities?
What is a Gorenstein surface?
What is a complete curve?
What is a Looijenga root?
What is the importance of \((-2){\hbox{-}}\)curves?
What is the Riemann-Hurwitz formula?
What is the adjunction formula?
What is Chen’s theorem?
What is the semistable reduction theorem?
What is a KSBA compactification?
What is the Lefschetz Hyperplane Theorem?
What is the Kummer surface associated to an abelian surface?
What is the ample cone?