Generalized Cayley's $Ω$-processes

Virtual residue and generalized Cayley- Bacharach Theorem

Using virtual residue, which is a generalization of Grothendieck residue, we generalized Cayley- Bacharach Theorem to the cases with positive dimensions.

Cayley K3曲面点Hilbert方案的Nef锥

Nef Cones of the Hilbert Schemes of Points on Generalized Cayley K3 Surfaces

本文研究了Cayley K3曲面点Hilbert方案的Nef锥，通过自同构和格论方法计算了S^{[2]}的Nef锥，并利用Bridgeland稳定性方法确定了更高阶Hilbert方案的Nef锥结构。

The generalized Cayley map from an algebraic group to its Lie algebra

Each infinitesimally faithful representation of a reductive complex connected algebraic group $G$ induces a dominant morphism $Φ$ from the group to its Lie algebra $\g$ by orthogonal projection in the endomorphism ring of the representation space. The map $Φ$ identifies the field $Q(G)$ of rational functions on $G$ with an algebraic extension of the field $Q(\g)$ of rational functions on $\g$. For the spin representation of $\on{Spin}(V)$ the map $Φ$ essentially coincides with the classical Cayley transform. In general, properties of $Φ$ are established and these properties are applied to deal…

Cayley forms and self dual varieties

Generalized Chow forms were introduced by Cayley for the case of 3-space, their zero set on the Grassmannian G(1,3) is either the set Z of lines touching a given space curve (the case of a `honest' Cayley form), or the set of lines tangent to a surface. Cayley gave some equations for F to be a generalized Cayley form, which should hold modulo the ideal generated by F and by the quadratic equation Q for G(1,3). Our main result is that F is a Cayley form if and only if Z = G(1,3) \cap {F=0} is equal to its dual variety. We also show that the variety of generalized Cayley forms is defined by quad…

CM Stability and the Generalized Futaki Invariant I

Based on the Cayley, Grothendieck, Knudsen Mumford theory of determinants we extend the CM polarization to the Hilbert scheme. We identify the weight of this refined line bundle with the generalized Futaki invariant of Donaldson. We are able to conclude that CM stability implies K-Stability. An application of the Grothendieck Riemann Roch Theorem shows that this refined sheaf is isomorphic to the CM polarization introduced by Tian in 1994 on any closed, simply connected base .