K-stability and birational models of moduli of quartic K3 surfaces

Abstract

We show that the K-moduli spaces of log Fano pairs $$({\mathbb {P}}^3, cS)$$ ( P 3 , c S ) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily–Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza–O’Grady’s prediction on the Hassett–Keel–Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $${\mathbb {P}}^3$$ P 3 .

Wall crossing for K‐moduli spaces of plane curves

“…(4) The wall crossing framework of this paper has been applied to the study of moduli of quartic K3 surfaces in [3,4]. In particular, the paper [4] verifies Laza-O'Grady's conjecture on the Hassett-Keel-Looijenga program for quartic K3 surfaces (see [85][86][87] for backgrounds). (5) For moduli of stable pairs in terms of Kollár-Shepherd-Barron-Alexeev, the wall crossing framework was established in [2].…”

“…6), there is only one wall crossing for K-moduli spaces given by the weighted blow-up PGIT In fact, in the degree 4 case, we can say more using Hyeon and Lee's results on the log MMP for moduli of genus three curves [54]; see Section 9.3.1. (ℙ 2 , 𝐴 11 -irreducible quintics) (ℙ (1,4,25), (𝑧 2 + 𝑥 2 𝑦 12 + 𝑥 10 g(𝑥, 𝑦) = 0))…”

“…As a result, the construction of our moduli spaces $$\mathcal {K}\mathcal {M}_{\chi _0, r,c}$$ is generalized to all log Fano pairs using purely algebraic methods. (2)The Yau–Tian–Donaldson conjecture for all (possibly singular) log Fano pairs is solved as a combination of [14, 80, 91, 98]. (3)The wall crossing framework from Section 3, except the local VGIT presentation from Section 3.7, is generalized to all log Fano pairs $$(X,D)$$ satisfying $$D\sim _{\mathbb {Q}} -r K_X$$ for some $$r\in \mathbb {Q}_{>0}$$ in [128] using purely algebraic methods. (4)The wall crossing framework of this paper has been applied to the study of moduli of quartic K3 surfaces in [3, 4]. In particular, the paper [4] verifies Laza–O'Grady's conjecture on the Hassett–Keel–Looijenga program for quartic K3 surfaces (see [85–87] for backgrounds). (5)For moduli of stable pairs in terms of Kollár–Shepherd–Barron–Alexeev, the wall crossing framework was established in [2].…”

K-Moduli of Curves on a Quadric Surface and K3 Surfaces

“…Remark 6.11. In a forthcoming work [7], we give a complete description of wall crossing for K-moduli compactifications of P 3 ,cS , where S ⊂ P 3 is a smooth degree 4 K3 surface. As an application, we prove Laza and O'Grady's conjecture [42,44] on birational models of moduli of degree 4 K3 surfaces.…”

Infinitesimal structure of log canonical thresholds

“…2 belongs to a finite set of positive real numbers and ˇ0 WD .P be general elements in jH j,C WD E \ H 1 \ H 2 \ H d 2 , and r WD b 1 " cŠ. Since .Y =X 3 x; B Y C E/ is "-plt, by[12, Theorem 3.10], rEj E is a Q-Cartier Weil divisor.…”