arXiv subjects

Baiting Xie

Publications and source records attributed to Baiting Xie.

7 recordsLinked to original sources

arXiv2026

Small-Subgroup Criteria for Liftability of Automorphism Groups of Smooth Hypersurfaces

In this paper, building on our previous Sylow criteria, we establish small-subgroup criteria of liftability and $F$-liftability for the linear automorphism group $G$ of smooth hypersurfaces $X$ over algebraically closed field of characteristic zero. When $\mathrm{dim} X = p-2$ for some odd prime $p$, we prove that ($F$-)liftability of any finite subgroup of $G$ can be tested on its $p$-subgroups of order at most $p^2$. When $X$ is a degree $p$ hypersurface of dimension $2p-2$, we prove that ($F$-)liftability of $G$ can be tested on all its $p$-subgroups of order at most $p^3$, which is sharp for $p\geq5$. When $p=3$, this bound improves to $9$, giving the corresponding criteria for smooth cubic fourfolds.

math.AG↗

arXiv2026

Monodromy Eigenvalues of Milnor Fibers for Line Arrangements

It is an important problem to know whether the monodromy on the cohomology of Milnor fibers associated to hyperplane arrangements is a combinatorial invariant. In this paper, we obtain a combinatorial vanishing criterion for certain eigenspaces of this algebraic monodromy in line arrangements. Combining with Hirzebruch inequality, which is a consequence of Bogomolov--Miyaoka--Yau inequality, we prove that for essential complex line arrangements, the eigenvalues of the monodromy have orders at most five. This is a partial progress toward Papadima--Suciu conjecture and proves Salvetti--Serventi connectivity conjecture. For essential complexified real line arrangements, the monodromy order is improved to at most four thanks to Shnurnikov's inequality. This confirms Papadima--Suciu conjecture for real line arrangements and also Yoshinaga's sharp pair conjecture.

math.AG↗

arXiv2026

Sylow Criteria for Liftability of Automorphism Groups of Smooth Hypersurfaces

In this paper, we first establish Sylow criteria for liftability of finite subgroups of the projective linear groups over arbitrary field. We also obtain Sylow criteria for $F$-liftability: if $F$ is a nonsingular polynomial of degree $d$ in $N$ variables over a field of characteristic zero, then a finite subgroup $G$ of $\mathrm{Lin}(F)$ is $F$-liftable if and only if for every prime $p$ dividing $\gcd(|G|,N,d)$, there exists a Sylow $p$-subgroup of $G$ that is $F$-liftable. Our proof combines restriction and corestriction in group cohomology with the reduction-to-Klein method.

math.AG↗

arXiv2026

Combinatorial Nonresonance Theorems for Hyperplane Arrangement Complements

We study the nonresonance phenomenon for complex rank-one local systems on complements of hyperplane arrangements. We refine the method of Cohen, Dimca, and Orlik and obtain a combinatorial sufficient condition for nonresonance. As an application, we strengthen a theorem of Bailet, Dimca, and Yoshinaga by removing one of its conditions. We also develop restriction and lifting techniques to prove a nonresonance theorem for line arrangements.

math.AG↗

arXiv2026

Log-concavity from enumerative geometry of planar curve singularities

We propose a log-concavity conjecture for BPS invariants arising in the enumerative geometry of planar curve singularities, identified with the local Euler obstructions of Severi strata in their versal deformations. We further extend this conjecture to ruling polynomials of Legendrian links and to E-polynomials of character varieties. We establish these conjectures for irreducible weighted-homogeneous singularities (torus knots) and for ADE singularities, and prove a multiplicative property for ruling polynomials compatible with log-concavity.

math.AG↗

arXiv2026

Homology of Local Systems on Real Line Arrangement Complements

We study the homology groups of the complement of a complexified real line arrangement with coefficients in complex rank-one local systems. Using Borel--Moore homology, we establish an algorithm computing their dimensions via the real figures of the arrangement. It enables us to give a new upper bound. We further consider the case where the arrangement contains a sharp pair and make partial progress on a conjecture proposed by Yoshinaga.

math.AG↗

arXiv2026

The{N/D}-Conjecture for Nonresonant Hyperplane Arrangements

This paper studies Bernstein--Sato polynomials $b_{f,0}$ for homogeneous polynomials $f$ of degree $d$ with $n$ variables. It is open to know when $-{n\over d}$ is a root of $b_{f,0}$. For essential indecomposable hyperplane arrangements, this is a conjecture by Budur, Mustaţă and Teitler and implies the strong topological monodromy conjecture for arrangements. Walther gave a sufficient condition that a certain differential form does not vanish in the top cohomology group of Milnor fiber. We use Walther's result to verify the $n\over d$-conjecture for weighted hyperplane arrangements satisfying the nonresonant condition.

math.AG↗