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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 weightm Γautomorphic forms. We let λ(Λ, Γ) := c1(L (Λ, Γ)); thus, λ(Λ, Γ) is a QCartier divisor class.” (Laza 和 O’Grady, 2019, p. 1663)
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.”
ample L or semiample or pseudoample or nef and big. But all primitive
smooth or ADE singularity
e.g.
K3 lattice obsidian K3 lattice
MMPforFdlt
“Theorem A. Let (X, F, B) be a Qfactorial projective Fdlt foliated triple such that F is algebraically integrable. Let A be an ample Rdivisor 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 semiample1. (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 QCartier, 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
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 $\mathcal{H}$
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
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
$rational \subset unirational \subset rationally connected \subset uniruled$
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.
Let $f: Y \dashrightarrow X$ be a rational dominant map, and $Y,X$ smooth projective varities, then $\kappa (Y) \geqslant \kappa (X)$
Remark: