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Hodge bundle-DINTFLHQ
Hodge bundle
“The Hodge bundle (or automorphic bundle), denoted by L (Λ, Γ), is a fundamental fractional (orbifold) line bundle on FΛ(Γ); it is defined as the quotient of OD + Λ (−1) by Γ, where OD+ Λ (−1) is the restriction to D+ Λ of the tautological line bundle on P(ΛC). We recall that L (Λ, Γ) extends to an ample fractional line bundle L ∗(Λ, Γ) on the Baily–Borel compactification FΛ(Γ)∗, and that the sections of mL ∗(Λ, Γ) are precisely the weight-m Γ-automorphic forms. We let λ(Λ, Γ) := c1(L (Λ, Γ)); thus, λ(Λ, Γ) is a Q-Cartier divisor class.” (Laza 和 O’Grady, 2019, p. 1663)
K3 lattice-ZHJBYNYC
K3 lattice
“second integral cohomology group of a K3 surface, endowed with the bilinear symmetric form given by cup product, is an even unimodular lattice of signature (3,19) and is unique up to isomorphism. Thus (see also §3.1 below for the notation), there is an isometric 2 32 isomorphism of H (X;Z) with the lattice A = H ©Eft . We call A the K3 lattice.”
K3Moduli-DLFQJ2XA
K3Moduli
Polarized K3 surfaces
ample L or semi-ample or pseudo-ample or nef and big. But all primitive
smooth or ADE singularity
polarized and morphism and GIT moduli
e.g.
Cohomology etc
K3 lattice obsidian K3 lattice
MMPforFdlt-EHKWJVZ2
MMPforFdlt
“Theorem A. Let (X, F, B) be a Q-factorial projective F-dlt foliated triple such that F is algebraically integrable. Let A be an ample R-divisor on X. Then: (1) The cone theorem, contraction theorem, and the existence of flips hold for (X, F, B). In particular, we can run a (KF + B)-MMP. (2) If KF + B + A is nef, then KF + B + A is semi-ample1. (3) If B ≥ A ≥ 0, then (X, F, B) has a good minimal model or a Mori fiber space. (4) If KF + B + A is Q-Cartier, then the canonical ring of KF + B + A, R(X, KF + B + A) = +∞ ⊕ m=0 H0(X, OX (⌊m(KF + B + A)⌋)), is finitely generated.” (Chen 等, 2023)
Period point in Hyperelliptic Unigonal divisor-SS4PXY2K
Period point in Hyperelliptic Unigonal divisor
Proposition 2.2.1. Let (X, L) be a quartic K3 surface. Then: (1) (X, L) is hyperelliptic if and only if Π(X, L) ∈ Hh(19); and (2) (X, L) is unigonal if and only if Π(X, L) ∈ Hu(19).
Stratification of $\mathcal{H}$-RWRLHEEQ
Stratification of $\mathcal{H}$
In between, we expect a series of flips, dictated by the structure of the preimage of Δ under the quotient map π : D → F . More precisely, let H := π−1(supp Δ); then H is a union of hyperplane sections of D, and hence is stratified by closed subsets, where a stratum is determined by the number of independent sheets (“independent sheets” means that their defining equations have linearly independent differentials) of H containing the general point of the stratum. The stratification of H induces a stratification of supp Δ, where the strata of supp Δ are indexed by the “number of sheets” (in D, not in F = Γ \D). Roughly speaking, Looijenga predicts that a stratum of supp Δ corresponding to k (at least) sheets meeting (in D) is flipped to a dimension k − 1 locus on the GIT side.
coarse moduli space for polarized K3-9ZSF59KR
coarse moduli space for polarized K3
The global Toreili theorem ([29], [11]) states that fl(h)/r(h) is a coarse moduli space for primitively polarized K3 surfaces of degree 2k
2
rationality
$rational \subset unirational \subset rationally connected \subset uniruled$
where uniruled if $\mathbb{P}^1\times Y \dashrightarrow X$ dominate.
Def: A variety $X$ is ratinally connected if it is proper and exists a variety $M$ and a rational map
$$ e:\mathbb{P}^1\times M \dashrightarrow X $$
s.t. the rational map
$$ \mathbb{P}^1\times \mathbb{P}^1 \times M \dashrightarrow X\times X $$
is dominant
An example for uniruled but not rationality connected:
$X = \mathbb{p}^1\times E$ with $g(E)\geqslant 1$. All rational curves in $X$ is vertical.
Exercise
Let $f: Y \dashrightarrow X$ be a rational dominant map, and $Y,X$ smooth projective varities, then $\kappa (Y) \geqslant \kappa (X)$
Remark: