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# 05-20-2022

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 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?

• 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 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 $${\mathbf{P}}^2$$?
• 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 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 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 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?

• What does it mean for $${\mathcal{L}}\in \operatorname{Pic}(X)$$ to be big and nef?
• What is the self-intersection $$({\mathcal{L}})^2$$?