K-moduli of curves on a quadric surface and K3 surfaces

Ascher, Kenneth; DeVleming, Kristin; Liu, Yuchen

doi:10.48550/arxiv.2006.06816

Cited by 7 publications

(29 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note that partial results toward Laza-O'Grady's prediction were obtained in [LO19,LO18]. In [LO21] the 18-dimensional case of their prediction was confirmed (see also [ADL20] for a different approach).…”

mentioning

confidence: 60%

“…The remainder of the birational transformations are flips, finally followed by a divisorial contraction F 18 (1 − ǫ) → M GIT (4,4) . In [ADL20], we show that the wall crossings resolving the period map p can be interpreted as wall-crossings in a suitable K-moduli space of log Fano pairs. Let K c denote the connected component of the moduli stack parametrizing K-semistable log Fano pairs admitting Q-Gorenstein smoothings to (P 1 × P 1 , cC) where C is a (4, 4) curve.…”

Section: Git Stratification Of Quartic Surfaces By Proposition 36 The...mentioning

confidence: 99%

“…This is the trickiest part of the proof. Our motivation comes from [ADL20] where we show that the K-moduli compactification K c of (P 1 × P 1 , cC) where C ∈ |O(4, 4)| is identical to the VGIT moduli space of slope t = 3c 2c+2 . By taking fiberwise double covers, we obtain a family of K-moduli spaces birational to H h .…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Ascher¹,

DeVleming²,

Liu³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We show that the K-moduli spaces of log Fano pairs (P 3 , cS) 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 P 3 .as an anti-canonical divisor. Then using a cone construction, a covering trick, and interpolation (see Section 5.1 for more detalis), we show that K 3c−1 4 admits a closed embedding into M K c for c > 1 3 whose image H h,c is a birational transform of H h (see Theorem 5.7). Then we construct K-polystable replacements of W i by first embedding all of P 3 , X h , P(1, 1, 2, 4) into P(1 4 , 2) as weighted hypersurfaces of degree two, and then finding a particular 1-PS (coming from VGIT in [LO21]) that degenerates (P 3 , cS) to a K-polystable pair in H h,c (see Theorem 5.10). Then we use deformation theory to classify exceptional loci after the walls. In particular, all K-moduli spaces M K c are isomorphic outside of the loci H h,c and H u,c . We give a complete description of all wall-crossings of M K c in Theorem 5.14. Finally, in Section 6 we prove the main theorems. We observe that F M K c is a birational contraction by Theorem 5.14. The upshot to showto use ampleness of log CM line bundles [XZ20], and to compute the variation of log CM line bundles which interpolate between the Hodge line bundle and the absolute CM line bundle (see (6.1)). Then we perform necessary birational modifications on M K c to obtain F(a, b) (see Definition 6.4), and show that the pushforward of λ + a 2 H h + b 2 H u is ample. 13 Theorem 4.19. The K-moduli spaces M K c has a wall at c = 9 13 . Moreover, we have (1) The wall crossing morphism φ − : M K 9 13 −ǫ → M K 9 13 replaces [(P 3 , T )] by [(X u , T 0 )], and is isomorphic near [(X u , T 0 )]. (2) The wall crossing morphism φ + : M K 9 13 +ǫ → M K 9 13 replaces [(X u , T 0 )] by the divisor H u, 9 13 +ǫ . (3) For any c ∈ ( 9 13 , 1), the birational map M K c M K 9 13 +ǫ is an isomorphism over a neighborhood of H u, 9 13 +ǫ . Proof. (1) Let U T := M GIT \ W 8 be an open neighborhood of [T ]. By (3.1), we know that any [S] ∈ U T \ {[T ]} is slc, thus (P 3 , cS) is K-stable for any c ∈ (0, 1). Since kst(P 3 , T ) = 9 13 by Theorem 4.3, there are open immersions U T ֒→ M K c when c ∈ (0, 9 13 ) and U T \ {[T ]} ֒→ M K c when c ∈ [ 9 13 , 1). Thus the map φ − : M K 9 13 −ǫ → M K 9 13is isomorphic on U T \ {[T ]}. On the other hand, we know that φ − ([(P 3 , T )]) = [(X u , T 0 )] by Proposition 4.7. By Corollary 4.17, we know thatis normal near [(X u , T 0 )] by Lemma 2.19. By Zariski's main theorem, we know that (φ − ) −1 ([(X u , T 0 )]) is connected, thus it has to be the singleton {[(P 3 , T )]}. Hence φ − is...

show abstract

mentioning

confidence: 60%

Section: Git Stratification Of Quartic Surfaces By Proposition 36 The...mentioning

confidence: 99%

See 1 more Smart Citation

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

Ascher¹,

DeVleming²,

Liu³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The case of smoothable del Pezzo surfaces has been extensively studied [22,24,25]. Moreover, K-moduli are understood for cubic 3-folds [20], cubic 4-folds [18], and for certain pairs (surface, curve) [5,6].…”

Section: Introductionmentioning

confidence: 99%

A 1-dimensional component of K-moduli of del Pezzo surfaces

Petracci¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Some cases for log Fano pairs have been worked out as well, see e.g. [Fuj17,GMGS18,ADL19,ADL20]. Despite these results, much less is known in higher dimensions.…”

Section: Introductionmentioning

confidence: 99%

K-stability of cubic fourfolds

Liu¹

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We prove that the K-moduli space of cubic fourfolds is identical to their GIT moduli space. More precisely, the K-(semi/poly)stability of cubic fourfolds coincide to the corresponding GIT stabilities, which was studied in detail by Laza. In particular, this implies that all smooth cubic fourfolds admit Kähler-Einstein metrics. Key ingredients are local volume estimates in dimension three due to Liu-Xu, and Ambro-Kawamata's non-vanishing theorem for Fano fourfolds.

show abstract

K-moduli of curves on a quadric surface and K3 surfaces

Cited by 7 publications

References 37 publications

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

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

A 1-dimensional component of K-moduli of del Pezzo surfaces

K-stability of cubic fourfolds

Contact Info

Product

Resources

About