Virtual fundamental classes, global normal cones and Fulton's canonical classes

A K-theoretic Fulton class

Fulton defined classes in the Chow group of a quasi-projective scheme $M$ which reduce to its Chern classes when $M$ is smooth. When $M$ has a perfect obstruction theory, Siebert gave a formula for its virtual cycle in terms of its total Fulton class. We describe K-theory classes on $M$ which reduce to the exterior algebra of differential forms when $M$ is smooth. When $M$ has a perfect obstruction theory, we give a formula for its K-theoretic virtual structure sheaf in terms of these classes.

Balanced normal cones and Fulton-MacPherson's intersection theory

Let X be a subscheme of a reduced scheme Y. Then Y has a flat "degeneration to the normal cone" C_X Y of X, and this degeneration plays a key step in Fulton and MacPherson's "basic construction" in intersection theory. The intersection product has a canonical refinement as a sum over the components of C_X Y, for X and Y depending on the given intersection problem. The cone C_X Y is usually not reduced, which leads to the appearance of multiplicities in intersection formulae. We describe a variant of this degeneration, due essentially to Samuel, Rees, and Nagata, in which Y flatly degenerates t…

虚拟类：导论与练习

Virtual classes: An introduction with exercises

本文介绍Behrend-Fantechi虚拟类，推广Fulton局部化顶部陈类，并推导具有全局分解的完美障碍理论的Deligne-Mumford栈上虚拟类的Siebert公式。

Integral points and effective cones of moduli spaces of stable maps

Consider the Fulton-MacPherson configuration space of $n$ points on $¶^1$, which is isomorphic to a certain moduli space of stable maps to $¶^1$. We compute the cone of effective ${\mathfrak S}_n$-invariant divisors on this space. This yields a geometric interpretation of known asymptotic formulas for the number of integral points of bounded height on compactifications of $\SL_2$ in the space of binary forms of degree $n\ge 3$.

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…

Schur类的正性性质

Positivity properties of Schur classes

本文将丰富和数值有效向量丛的Schur类正性结果推广到纯代数环境，证明具有Kähler包的射影丛环满足Hodge-Riemann关系，并由此推出拟阵Schur系数非负。