Multigraded Cayley-Chow forms

Multigraded Hilbert Schemes

We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely applicable, it provides explicit equations, and it allows us to prove a range of new results, including Bayer's conjecture on equations defining Grothendieck's classical Hilbert scheme and the construction of a Chow morphism for toric Hilbert schemes.

多分级Hurwitz形式

Multigraded Hurwitz forms

本文提出多分级Hurwitz形式的概念，研究其次数并应用于消元论，具有从纳什均衡到费曼积分的广泛应用

The Chow ring of the Cayley plane

We give a full description of the Chow ring of the complex Cayley plane, the simplest of the exceptional flag varieties. We describe explicitely the most interesting of its Schubert varieties and compute their intersection products. Translating our results in the Borel presentation, i.e. in terms of Weyl group invariants, we are able to compute the degree of the variety of reductions $Y_8$ introduced in our related preprint math.AG/0306328.

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…