Segre Class of Schemes with Regularly Embedded Components
Guanxi Li
TL;DR
This work extends Fulton’s Residual Intersection framework to Segre classes for schemes with regularly embedded components, introducing the $\odot$ residual product and the $\mathscr{L}$-tensored Segre class to express how components and their intersections contribute to the Segre class of a union. The authors derive a transverse-intersection formula for $s(W,Z)$ that expresses it in terms of the Segre classes of the components and their normal bundles, and they generalize this to non-transverse cases using a deformation-to-the-normal-cone approach, yielding a comprehensive residual-intersection expression involving $N_{X\cap Y}X$, $N_{X\cap Y}Y$, and $N_{X\cap Y}Z$. Central to the results are explicit blowup analyses and Chern-class computations of the normal bundles after blowups, organized through an auxiliary $Q$-polynomial and the $\odot$ operation. The methods provide precise, implementable formulas that relate the Segre class of a union to component-wise Segre data, with potential applications in intersection theory and enumerative geometry.
Abstract
We generalize Fulton's Residual Intersection Theorem for the Segre class and express the Segre classes of schemes with regularly embedded components in terms of the Chern classes of the normal bundles to the components and their intersections. More specifically, we provide formulas for the following situations: when the components of the scheme intersect transversely and when the ideal sheaf of the scheme, after the blowup along a component, is the product of the ideal sheaves of the exceptional divisor and the residual scheme.