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Fei Si

Publications and source records attributed to Fei Si.

9 recordsLinked to original sources
arXiv2025

Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces

In this paper, we develop a unified approach to study the intersection Betti numbers of moduli spaces of one-dimensional semistable sheaves on smooth projective surfaces. Assuming the irreducibility of such moduli spaces, we prove that their intersection Betti numbers in a certain range of degrees coincide with the stable Betti numbers of Hilbert schemes of points. As an application, for surfaces with nef anticanonical divisor, we show that these intersection Betti numbers stabilize in each fixed degree, which fits into the broader context of stable cohomology for moduli spaces of sheaves; if in addition the moduli spaces are smooth, we also prove a refined stabilization result on perverse Hodge numbers.

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arXiv2025

On the Picard numbers of moduli spaces of one-dimensional sheaves on surfaces

Motivated by asymptotic phenomena of moduli spaces of higher rank stable sheaves on algebraic surfaces, we study the Picard number of the moduli space of one-dimensional stable sheaves supported in a sufficiently positive divisor class on a surface. We give an asymptotic lower bound of the Picard number in general. In some special cases, we show that this lower bound is attained based on the geometry of moduli spaces of stable pairs and relative Hilbert schemes of points. Additionally, we discuss several related questions and provide examples where the asymptotic irreducibility of the moduli space fails, highlighting a notable distinction from the higher rank case.

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arXiv2024

Asymptotic Behaviors of Moduli of One-dimensional Sheaves on Surfaces

In this paper, we study the asymptotic behaviors of the Betti numbers and Picard numbers of the moduli space $M_{\beta,\chi}$ of one-dimensional sheaves supported in a curve class $\beta$ on $S$ with Euler characteristic $\chi$. We determine the intersection cohomology Betti numbers of $M_{\beta,\chi}$ when $S$ is a del Pezzo surface and $\beta$ is sufficiently positive. As an application, we formulate a $P = C$ conjecture regarding the refined BPS invariants for local del Pezzo surfaces.

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arXiv2024

Cohomological stabilization, perverse filtrations, and refined BPS invariants for del Pezzo surfaces

We prove an asymptotic product formula for the refined BPS invariants associated with a local del Pezzo surface. Our formula governs the cohomological stabilization of the perverse filtration on the intersection cohomology of the moduli space of 1-dimensional semistable sheaves on a del Pezzo surface. Combined with the theory of Fourier transform of Maulik--Shen--Yin, we show that the perverse filtration matches asymptotically with the Chern filtration defined via tautological classes. In the case of the projective plane, our results resolve conjectures of Kononov--Pi--Shen.

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arXiv2023

Birational geometry of moduli space of del Pezzo pairs

In this paper, we investigate the geometry of the moduli space $P_d$ of degree $d$ smooth del Pezzo pairs, which consists of a smooth del Pezzo surface $X$ of degree $d$ and a smooth curve $C \sim -2K_X$. More precisely, we study the compactifications of $P_d$ from both Hodge-theoretic and geometric invariant theoretical (GIT) perspectives. We obtain the class numbers of the Baily-Borel compactification $P_d^\ast$ for $P_d$, which is an important step toward establishing the Hassett-Keel-Looijenga program for $P_d$. If $d=8$, $P_d$ has two connected components. For the component parametrizing del Pezzo pairs $(Bl_p \PP^2, C)$, we propose the Hassett-Keel-Looijenga models $\cF(s)=\proj R(\cF,\Delta(s) )$ via the section rings of certain $\bQ$-line bundles $\Delta(s)$ on the locally symmetric variety $\cF$. These models are expected to connect different birational models of the moduli space arising from K-moduli theory. By constructing an arithmetic stratification on $\cF$ and computing the pullback of $\Delta(s)$ on these strata, we give arithmetic predictions for the wall-crossing of $\cF(s)$ as $s\in [0,1]$ varies. This work parallels that of Laza-O'Grady \cite{LO19, LaO18}.

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arXiv2023

Two results regarding the variation of K-moduli

In this note, we prove two results regarding the variation of K-moduli. The first one reveals the relationship between the chamber decomposition for K-semistable domains and the variation of GIT. The second one presents the relationship between the K-moduli generically parametrizing K-semistable smooth Fano complete intersections of the form $S_1\cap...\cap S_k$ and the K-moduli generically parametrizing K-semistable log Fano manifolds of the form $(\mathbb{P}^n, \sum_{j=1}^kx_jS_j)$, where $x_j\in (0,1)\cap \mathbb{Q}$ and $S_j\subset \mathbb{P}^n$ is a hypersurface of degree $d_j$ for each $1\leq j\leq k$.

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arXiv2023

K moduli of log del Pezzo pairs

We establish the full explicit wall-crossing for K-moduli space $\overline{P}^K_c$ of degree $8$ del Pezzo pairs $(X,cC)$ where generically $X \cong \bbF_1$ and $C \sim -2K_X$. We also show K-moduli spaces $\overline{P}^K_c$ coincide with Hassett-Keel-Looijenga(HKL) models $\cF(s)$ of a $18$-dimensional locally symmetric spaces associated to the lattice $E_8\oplus U^2\oplus E_7\oplus A_1$.

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arXiv2021

Cohomology of moduli space of cubic fourfolds I

In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan's blowup and Laza's GIT construction. More precisely, we obtain the Betti numbers of Kirwan's resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space.

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