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05202022
Let \(X\) be a K3 surface.
Basics
 What is a K3?
 What is \(\chi(X)\)?
 What is \(c_1(X)\)? \(c_2(X)\)?
 What is the lattice decomposition of \(H^2(X; {\mathbf{Z}})\)? 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 local Torelli?
 What is global Torelli?
 What is a double curve?
 What is a triple point?
 What is a proper variety?
 What is semistability?
 What is a Weil divisor?
 What is \({\mathcal{O}}(n), {\mathcal{O}}(n)\)?
 What are extended Dynkin diagrams?
 What is the degree of a K3 surface?
 What is a dominant morphism?
Reading
 What is a minimal resolution?
 What is a Kulikov surface?
 What is a Looijenga semitoroidal compactification?
 What is a polarized K3?
 What is a toroidal compactification?

What is \(F_{2d}\)?
 What is the BaileyBorel compactification?
 What is an admissible fan?
 What are the associated toroidal compactifications?
 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 YauZaslow formula?
 Why is a generic K3 a double branched cover of \({\mathbf{P}}^2\)?
 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 NeronSeveri?
 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 GaussManin 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 latticepolarized K3?
 What is a Weyl chamber?
 What is an Mquasipolarized K3 surface?
 What is an Mquasipolarized 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 is a hyperbolic lattice?
 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 EnriquesKodaira 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 dsemistability?
 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 PicardLefschetz 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 NoetherLefschetz 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 RiemannHurwitz 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?
Reading 2
 What does it mean for \({\mathcal{L}}\in \operatorname{Pic}(X)\) to be big and nef?
 What is the selfintersection \(({\mathcal{L}})^2\)?
 What is a semistable point?
 What is a VHS?