arXiv subjects

Junyan Zhao

Publications and source records attributed to Junyan Zhao.

17 recordsLinked to original sources

arXiv2026

Degenerations of elliptic quartics and K-moduli of Fano threefolds

We study the K-moduli space of Fano threefolds obtained by first blowing up $\mathbb{P}^3$ along an elliptic quartic curve $C$ and then blowing up a fiber $\ell_p$ of the exceptional divisor $E \to C$. We prove that this K-moduli space is isomorphic to a VGIT quotient parametrizing pairs $(C,p)$, with linearization induced by the CM line bundle. In particular, we classify all K-(semi/poly)stable members of this deformation family. The main new ingredients in the proof include the geometry of the Hilbert scheme of elliptic quartic curves, deformation theory of Fano--K3 pairs, and optimal bounds on the local volumes of threefold singularities.

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arXiv2026

Projective moduli of log Calabi--Yau fibrations over curves

We introduce a new stability condition for log Calabi--Yau fibrations over curves. We prove that it gives rise to a proper Deligne--Mumford stack with a projective coarse moduli space, whose boundary still parametrizes flat fibrations over curves.

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arXiv2026

Deformations of fibered Calabi--Yau varieties

Koll\'{a}r showed that small deformations of elliptically fibered smooth $K$-torsion varieties with $H^2(X,\mathcal{O}_X)=0$ remain elliptically fibered. We extend this result to any fibered smooth $K$-torsion variety $X$ with $H^2(X,\mathcal{O}_X)=0$, using Hodge theoretic techniques and the $T^1$-lifting criterion of Kawamata--Ran. More generally, our strategy implies that even without the cohomological assumption, small deformations of a semiample line bundle on a smooth $K$-torsion variety remain semiample up to homological equivalence.

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arXiv2026

The boundary of K-moduli of prime Fano threefolds of genus twelve

We study the K-moduli stack of prime Fano threefolds of genus twelve, known as $V_{22}$. We prove that its boundary, which parametrizes singular members, is purely divisorial and consists of four irreducible components corresponding to the four families of Prokhorov's one-nodal $V_{22}$. A key ingredient is a modular relation between Fano threefolds $X$ and their anticanonical K3 surfaces $S$. We prove that the forgetful morphism from the moduli of Fano--K3 pairs $(X,S)$ where $X$ is a K-semistable degeneration of $V_{22}$ to the moduli space of genus $12$ polarized K3 surfaces $(S,{-K_X}|_S)$ is an open immersion. In particular, the K-moduli of $V_{22}$ is governed by the moduli of their anticanonical K3 surfaces, providing a modular realization of Mukai's philosophy. Along the way, we develop a general deformation framework for Fano threefolds of large volume, which may be useful beyond the study of K-moduli.

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arXiv2025

Moduli of surfaces fibered in (log) Calabi-Yau pairs II: elliptic surfaces

This paper continues the study initiated in [ISZ25] on the moduli of surfaces admitting lc-trivial fibrations. Using the techniques developed in [ISZ25], we (1) provide a classification of the surfaces appearing on the boundary of the KSBA-moduli space of elliptic surfaces with a bisection (2) recover the results of a series of papers on the moduli stacks of elliptic surfaces with a section [AB22, Inc20, Bru15]. Notably, our proof of (2) avoids the use of explicit steps of an MMP, such as the "La Nave flip" from [LN02], which plays a central role in [AB22,Inc20]. As an application, we compactify the moduli stack of hyperelliptic K3 surfaces.

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arXiv2025

Moduli of surfaces fibered in log Calabi-Yau pairs

We study the moduli spaces of surface pairs $(X,D)$ admitting a log Calabi--Yau fibration $(X,D) \to C$. We develop a series of results on stable reduction and apply them to give an explicit description of the boundary of the KSBA compactification. Three interesting cases where our results apply are: (1) divisors on $\mathbb{P}^1 \times \mathbb{P}^1$ of bidegree $(2n,m)$; (2) K3 surfaces which map $2:1$ to $\mathbb{F}_n$, with $X=\mathbb{F}_n$ and $D$ the ramification locus, or (3) elliptic surfaces with either a section or a bisection. The main tools employed are stable quasimaps, the canonical bundle formula, and the minimal model program.

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arXiv2025

On K-stability of Fano's last Fanos

We study K-stability of smooth Fano threefolds of Picard rank $2$ and degree $22$ which can be obtained by blowing up a smooth complete intersection of two quadrics in $\mathbb{P}^5$ along a conic. We also describe the automorphism groups of these threefolds.

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arXiv2024

K-stability of Thaddeus' moduli of stable bundle pairs on genus two curves

The moduli space of bundle stable pairs $\overline{M}_C(2,\Lambda)$ on a smooth projective curve $C$, introduced by Thaddeus, is a smooth Fano variety of Picard rank two. Focusing on the genus two case, we show that its K-moduli space is isomorphic to a GIT moduli of lines in quartic del Pezzo threefolds. Additionally, we construct a natural forgetful morphism from the K-moduli of $\overline{M}_C(2,\Lambda)$ to that of the moduli spaces of stable vector bundles $\overline{N}_C(2,\Lambda)$. In particular, Thaddeus' moduli spaces for genus two curves are all K-stable.

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arXiv2024

Moduli of elliptic surfaces of Kodaira dimension one fibered over rational curves

In this article, we construct an infinite sequence of irreducible components of Koll\'{a}r--Shepherd-Barron (KSB-) moduli spaces of surfaces of arbitrarily large volumes, and describe the boundary of each component completely. Moreover, we describe the stable reduction steps in finding the KSB-limits in an explicit combinatorial way. Our main approach is to study the moduli spaces of elliptic surfaces with Kodaira dimension one, fibered over rational curves, using the techniques of wall-crossing for KSBA moduli and twisted stable maps.

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arXiv2024

K-moduli of log del Pezzo pairs and variations of GIT

We study the K-moduli of log del Pezzo pairs formed by a del Pezzo surface of degree $d$ and an anti-canonical divisor. These moduli spaces naturally depend on one parameter, providing a natural problem in variations of K-moduli spaces. For degrees 2, 3, 4, we establish an isomorphism between the K-moduli spaces and variations of Geometric Invariant Theory compactifications, which generalizes the isomorphisms in the absolute cases established by Odaka--Spotti--Sun and Mabuchi--Mukai.

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arXiv2024

Curves on complete intersections and measures of irrationality

We study the minimal degrees and gonalities of curves on complete intersections. We prove that the degree of any curve on a general complete intersection $X \subseteq \mathbb{P}^N$ of large multidegree is bounded from below by the degree of $X$. As an application, we answer a problem of Bastianelli--De Poi--Ein--Lazarsfeld--Ullery on measures of irrationality for complete intersections.

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arXiv2024

K-moduli of Fano threefolds and genus four curves

In this article, we study the K-moduli space of Fano threefolds obtained by blowing up $\mathbb{P}^3$ along $(2,3)$-complete intersection curves. This K-moduli space is a two-step birational modification of the GIT moduli space of $(3,3)$-curves on $\mathbb{P}^1 \times \mathbb{P}^1$. As an application, we show that our K-moduli space appears as one model of the Hassett--Keel program for $\overline{M}_4$. In particular, we classify all K-(semi/poly)stable members in this deformation family of Fano varieties. We follow the moduli continuity method with moduli of lattice-polarized K3 surfaces, general elephants and Sarkisov links as new ingredients.

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arXiv2023

Moduli of Genus Six Curves and K-stability

The K-moduli theory provides a different compactification of moduli spaces of curves. As a general genus six curve can be canonically embedded into the smooth quintic del Pezzo surface, we study in this paper the K-moduli spaces $\overline{M}^K(c)$ of the quintic log Fano pairs. We classify the strata of genus six curves $C$ appearing in the K-moduli by explicitly describing the wall-crossing structure. The K-moduli spaces interpolate between two birational moduli spaces constructed by GIT and moduli of K3 surfaces via Hodge theory.

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arXiv2023

The Moduli Space of Genus Six Curves and K-stability: VGIT and the Hassett-Keel Program

A general curve $C$ of genus six is canonically embedded into the smooth del Pezzo surface $Σ\subseteq \mathbb{P}^1 \times \mathbb{P}^2$ of degree $5$ as a divisor in the class $\mathcal{O}_Σ(2,2)$. In this article, we study the variation of geometric invariant theory (VGIT) for such pairs $(Σ,C)$, and relate the VGIT moduli spaces to the K-moduli of pairs $(Σ,C)$ and the Hassett-Keel program for moduli of genus six curves. We prove that the K-moduli spaces ${\overline{M}}^{K}(c)$ give the final several steps in the Hassett-Keel program for ${\overline{M}}_6$.

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arXiv2023

Compactifications of moduli of del Pezzo surfaces via line arrangement and K-stability

In this paper, we study compactifications of the moduli of smooth del Pezzo surfaces using K-stability and the line arrangement. We construct K-moduli of log del Pezzo pairs with sum of lines as boundary divisors, and prove that for $d=2,3,4$, these K-moduli of pairs are isomorphic to the K-moduli spaces of del Pezzo surfaces. For $d=1$, we prove that they are different by exhibiting some walls.

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arXiv2022

Moduli Spaces of Sheaves on General Blow-ups of $\mathbb{P}^2$

Let $X$ be the blow-up of $\mathbb{P}^2$ along $m$ general points, and $A=H-\sum \varepsilon_iE_i$ be a generic polarization with $0<\varepsilon_i\ll1$. We classify the Chern characters which satisfy the weak Brill-Noether property, i.e. a general sheaf in $M_A({\bf{v}})$, the moduli space of slope stable sheaves with Chern character ${\bf{v}}$, has at most one non-zero cohomology. We further give a necessary and sufficient condition for the existence of stable sheaves. Our strategy is to specialize to the case when the $m$ points are collinear.

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arXiv2022

$L^2$ Schrödinger maximal estimates associated with finite type phases in $\mathbb{R}^2$

In this paper, we establish Schrödinger maximal estimates associated with the finite type phases \begin{equation*} ϕ(ξ_1,ξ_2):=ξ^m_1+ξ^m_2,\;(ξ_1,ξ_2)\in [0,1]^2, \end{equation*} where $m \geq 4$ is an even number. Following [12], we prove an $L^2$ fractal restriction estimate associated with the surfaces \begin{equation*} F^2_m:=\{(ξ_1,ξ_2,ϕ(ξ_1,ξ_2)):\;(ξ_1,ξ_2)\in [0,1]^2\} \end{equation*} as the main result, which also gives results on the average Fourier decay of fractal measures associated with these surfaces. The key ingredients of the proof include the rescaling technique from [16], Bourgain-Demeter's $\ell^2$ decoupling inequality, the reduction of dimension arguments from [17] and induction on scales.

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