05-20-2022

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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 \(H^2(X; {\mathbb{Z}})\)? 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 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 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 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?

Reading 2

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