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Bochao Kong

Publications and source records attributed to Bochao Kong.

6 recordsLinked to original sources
arXiv2026

Tautological Pushforwards of Hilbert Schemes of Points on Curves and Surfaces

Ellingsrud, Göttsche, and Lehn proved a remarkable universality result for tautological integrals on Hilbert schemes of points on surfaces. We prove a relative form: over a base $B$, the tautological pushforwards are governed by universal power series paired with relative $κ$-classes. As applications, we focus on Segre classes on families of curves and surfaces. For curve families, we determine all universal coefficients for total Segre pushforwards. These formulas answer a question of Oprea--Pandharipande, and the resulting recursions are surprisingly related to monotone Hurwitz numbers. For surface families, motivated by Marian--Oprea--Pandharipande's work on Lehn's conjecture, we obtain a codimension-one relative form of the conjecture, with explicit formulas for every universal series.

math.AG↗
arXiv2026

A counterexample to a global-dimension bound for weighted projective lines

We observe that standard derived equivalences give a counterexample to a conjecture of Kalck on global dimension for weighted projective lines. For the root stack $X=\mathbb{P}^1\langle \infty,0,1;2,3,3\rangle$ we exhibit a $13$-dimensional radical-square-zero algebra $A$ such that $$ D^b(\mathrm{coh}X)\simeq D^b(\mathrm{mod}A), \qquad \mathrm{gldim}A=4>3. $$

math.AG↗
arXiv2026

Positive Scalar Curvature and Volume Growth

For a complete Riemannian manifold with nonnegative Ricci curvature, we prove two sharp volume growth order estimates, thereby resolve a conjecture of Gromov in 1986. There first is that a uniform deficit in the volume of unit balls, an analog of positive macroscopic scalar curvature, forces codimension one volume growth, and the second one is that a uniformly positive scalar curvature lower bound forces codimension two growth known as the codimension two volume growth conjecture.

math.DG↗
arXiv2026

To cover a permutohedron

The permutohedron $P_n$ of order $n$ is a polytope embedded in $\mathbb{R}^n$ whose vertex coordinates are permutations of the first $n$ natural numbers. It is obvious that $P_n$ lies on the hyperplane $H_n$ consisting of points whose coordinates sum up to $n(n+1)/2$. We prove that if the vertices of $P_n$ are contained in the union of $m$ affine hyperplanes different from $H_n$, then $m\geq n$ when $n \geq 3$ is odd, and $m \geq n-1$ when $n \geq 4$ is even. This result has been established by Pawlowski in a more general form. Our proof is shorter, rather different, and gives an algebraic criterion for a non-standard permutohedron generated by $n$ distinct real numbers to require at least $n$ non-trivial hyperplanes to cover its vertices.

math.CO↗
arXiv2021

The Chow rings of moduli spaces of elliptic surfaces over $\mathbb{P}^1$

Let $E_N$ denote the coarse moduli space of smooth elliptic surfaces over $\mathbb{P}^1$ with fundamental invariant $N$. We compute the Chow ring $A^*(E_N)$ for $N\geq 2$. For each $N\geq 2$, $A^*(E_N)$ is Gorenstein with socle in codimension $16$, which is surprising in light of the fact that the dimension of $E_N$ is $10N-2$. As an application, we show that the maximal dimension of a complete subvariety of $E_N$ is $16$. When $N=2$, the corresponding elliptic surfaces are K3 surfaces polarized by a hyperbolic lattice $U$. We show that the generators for $A^*(E_2)$ are tautological classes on the moduli space $\mathcal{F}_{U}$ of $U$-polarized K3 surfaces, which provides evidence for a conjecture of Oprea and Pandharipande on the tautological rings of moduli spaces of lattice polarized K3 surfaces.

math.AG↗