Extremality of Rational Tails Boundary Strata in $\overline{\mathcal{M}}_{g,n}$

Degenerations and multiplicity-free formulas for products of $ψ$ and $ω$ classes on $\overline{M}_{0,n}$

We consider products of $ψ$ classes and products of $ω$ classes on $\overline{M}_{0,n+3}$. For each product, we construct a flat family of subschemes of $\overline{M}_{0,n+3}$ whose general fiber is a complete intersection representing the product, and whose special fiber is a generically reduced union of boundary strata. Our construction is built up inductively as a sequence of one-parameter degenerations, using an explicit parametrized collection of hyperplane sections. Combinatorially, our construction expresses each product as a positive, multiplicity-free sum of classes of boundary strata…

On the effective cone of $\overline{\mathcal{M}}_{g,n}$

For every $g\geq 2$ and $n\geq g+1$ we exhibit infinitely many extremal effective divisors in $\overline{\mathcal{M}}_{g,n}$ coming from the strata of abelian differentials.

On the cone of effective 2-cycles on $\overline{M}_{0,7}$

Fulton's question about effective $k$-cycles on $\overline{M}_{0,n}$ for $1<k<n-4$ can be answered negatively by appropriately lifting to $\overline{M}_{0,n}$ the Keel-Vermeire divisors on $\overline{M}_{0,k+1}$. In this paper we focus on the case of $2$-cycles on $\overline{M}_{0,7}$, and we prove that the $2$-dimensional boundary strata together with the lifts of the Keel-Vermeire divisors are not enough to generate the cone of effective $2$-cycles. We do this by providing examples of effective $2$-cycles on $\overline{M}_{0,7}$ that cannot be written as an effective combination of the afore…

$k$-differentials on curves and rigid cycles in moduli space

For $g\geq2$, $j=1,\dots,g$ and $n\geq g+j$ we exhibit infinitely many new rigid and extremal effective codimension $j$ cycles in $\overline{\mathcal{M}}_{g,n}$ from the strata of quadratic differentials and projections of these strata under forgetful morphisms and show the same holds for $k$-differentials with $k\geq 3$ if the strata are irreducible. We compute the class of the divisors in the case of quadratic differentials which contain the first known examples of effective divisors on $\overline{\mathcal{M}}_{g,n}$ with negative $ψ_i$ coefficients.

Effective divisors in $\overline{\mathcal{M}}_{g,n}$ from abelian differentials

We compute many new classes of effective divisors in $\overline{\mathcal{M}}_{g,n}$ coming from the strata of abelian differentials and efficiently reproduce many known results obtained by alternate methods. Our method utilises maps between moduli spaces and the degeneration of abelian differentials.

Effective divisors in the projectivized Hodge bundle

We compute the class of the closure of the locus of canonical divisors in the projectivization of the Hodge bundle $\mathbb{P}\overline{\mathcal{H}}_g$ over $\overline{\mathcal{M}}_g$ which have a zero at a Weierstrass point. We also show that the strata of canonical and bicanonical divisors with a double zero span extremal rays of the respective pseudoeffective cones.

亏格0中d-微分地层关联簇紧化

The Incidence Variety Compactification of strata of d-differentials in genus 0

本文构造了亏格0中d-微分地层射影化的关联簇紧化，证明其同构于模空间沿特定理想层的爆破，并给出重言线丛的显式除子代表，为计算球面固定锥角平坦度量体积提供基础。

Hurwitz空间的边界分层

Boundary stratifications of Hurwitz spaces

本研究建立了Hurwitz空间紧化的不可约边界层与装饰树的组合对应关系，由此定义了加细的tropical Hurwitz空间并给出计算实现，为复动力系统等领域提供了新的组合与几何工具。

奇点的Ekedahl-Oort层与Oort猜想

Supersingular Ekedahl-Oort strata and Oort's conjecture

本文证明了在特定条件下，奇点Ekedahl-Oort层的几何通用成员的自同构群为$\{ \pm 1\}$，并验证了Oort猜想在偶数维和素数$p\geq 5$时的正确性。

$\mathcal{M}_{1,n}$的首个更高Chow群对于$n\leq 4$

The first higher Chow groups of $\mathcal{M}_{1,n}$ for $n\leq 4$

本文计算了$\mathcal{M}_{1,n}$对于$n\leq 4$的更高Chow群，并给出了积分Chow环的新证明及边界strata类的公式。

通用紧化雅可比：上同调不变性与边界组合学

Universal compactified Jacobians: cohomological invariance and boundary combinatorics

本文通过直接组合方法证明了紧化雅可比的上同调与参数d和稳定性条件σ无关，同时附录包含关于雅可比模的等价性和K群中类的非等价性结果。

等剩余纤维化与带点稳定曲线模空间的双有理几何

Isoresidual fibrations and the birational geometry of moduli of pointed stable curves

本研究针对带点稳定曲线模空间$\u0001overline{M}_{0,n}$的双有理几何问题，通过亚纯微分分层映射构造非多面极端射线，证明n≥8时其除子伪有效锥非多面，且这类空间不是Mori Dream Space。

极化阿贝尔簇何时由其 $p$-可除群确定？

When is a polarised abelian variety determined by its $\boldsymbol{p}$-divisible group?

该研究聚焦亏格$g$的Siegel模簇及其超奇轨迹，通过转化为类数为一问题，明确了超奇轨迹不可约的条件与极化阿贝尔簇由其极化$p$-可除群唯一确定的情形，求解用到质量公式、自同构群及$g=4$时Ekedahl-Oort层分析。