Shadows and intersections: stability and new proofs

Faber相交数猜想的一个新证明

A new proof of Faber's intersection number conjecture

该研究提出了曲线模空间$\bar{M}_g$的tautological环中Faber相交数猜想的新证明，证明基于双分歧循环的直接几何与组合计算，为该猜想提供了新的推导路径。

A local version of Katona's intersection theorem

Katona's intersection theorem states that every intersecting family $\mathcal F\subseteq[n]^{(k)}$ satisfies $\vert\partial\mathcal F\vert\geq\vert\mathcal F\vert$, where $\partial\mathcal F=\{F\setminus x:x\in F\in\mathcal F\}$ is the shadow of $\mathcal F$. Frankl conjectured that for $n>2k$ and every intersecting family $\mathcal F\subseteq [n]^{(k)}$, there is some $i\in[n]$ such that $\vert \partial \mathcal F(i)\vert\geq \vert\mathcal F(i)\vert$, where $\mathcal F(i)=\{F\setminus i:i\in F\in\mathcal F\}$ is the link of $\mathcal F$ at $i$. Here, we prove this conjecture in a very strong …

A short proof of Rudnev's point-plane incidence bound

In this note we give a shortened proof of a theorem of Rudnev, which bounds the number of incidences between points and planes over an arbitrary field. Rudnev's proof uses a map that goes via the four-dimensional Klein quadric to a three-dimensional space, where it applies a bound of Guth and Katz on intersection points of lines. We describe a simple geometric map that directly sends point-plane incidences to line-line intersections in space, allowing us to reprove Rudnev's theorem with fewer technicalities.

Matroid intersection, base packing and base covering for infinite matroids

As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and matroid intersection and union. We show that several of these conjectures are equivalent, so that each gives a perspective on the same central problem of infinite matroid theory. For finite matroids, these equivalences give new and simpler proofs for the finite theorems corresponding to these conjectures. This new point of view also allows us to extend, and…

A combinatorial proof of the smoothness of catalecticant schemes associated to complete intersections

For zero-dimensional complete intersections with homogeneous ideal generators of equal degrees over an algebraically closed field of characteristic zero, we give a combinatorial proof of the smoothness of the corresponding catalecticant schemes along an open subset of a particular irreducible component.

强弦图作为树的交图（Farber 证明的重述）

Strongly chordal graphs as intersection graphs of trees (Farber's proof revisited)

本文重述了 Farber 关于强弦图可表示为加权树的相容子树交图的未发表证明，并给出一个直接将相容子树转化为强消去序的替代充分性证明。

On the Hilton-Spencer intersection theorems for unions of cycles

A family $\mathcal{A}$ of sets is said to be intersecting if every two sets in $\mathcal{A}$ intersect. An intersecting family is said to be \emph{trivial} it its sets have a common element. A graph $G$ is said to be $r$-EKR if at least one of the largest intersecting families of independent $r$-element sets of $G$ is trivial. Let $α(G)$ and $ω(G)$ denote the independence number and the clique number of $G$, respectively. Hilton and Spencer recently showed that if $G$ is the vertex-disjoint union of a cycle ${_*C}$ raised to the power $k^*$ and $s$ cycles ${_1C}, \dots, {_sC}$ raised to the po…

Hodge指标定理阴影下的正性

Positivity in the shadow of Hodge index theorem

本文在紧致Kähler流形上利用Hodge指标定理研究Lorentzian类，刻画Lefschetz型算子的核面，为Teissier比例性问题提供新方法，并建立Brunn-Minkowski不等式极值与定量稳定性估计。

A short proof of the lambda_g-conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves

We give a short and direct proof of the $λ_g$-Conjecture. The approach is through the Ekedahl-Lando-Shapiro-Vainshtein theorem, which establishes the ``polynomiality'' of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of Gromov-Witten theory. We briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures.

Intersection patterns of finite sets and of convex sets

The main result is a common generalization of results on lower bounds for the chromatic number of r-uniform hypergraphs and some of the major theorems in Tverberg-type theory, which is concerned with the intersection pattern of faces in a simplicial complex when continuously mapped to Euclidean space. As an application we get a simple proof of a generalization of a result of Kriz for certain parameters. This specializes to a short and simple proof of Kneser's conjecture. Moreover, combining this result with recent work of Mabillard and Wagner we show that the existence of certain equivariant m…