Need references on moduli of subvarieties of given dimension and degree of the projective space P^N

Question

Hello,

I need to understand the moduli of subvarieties of given dimension and degree of the projective space P^N. Is there any good place to look into? Thank you.

Answer 1

Before talking of moduli, which certainly is a very good way to treat the set of subvarieties of given dimension and degree of a fixed projective space, you can try to understand Chow varieties which is exactly the algebraic variety whose points are all cycles of a given projective space of given dimension and degree. This is easier because we have coordinates called the Chow coordinates. This is done in J. Harris' book "Algebraic geometry: a first course" GTM springer, here you will find a very nice chapter on parameter spaces and moduli spaces (lecture 21). In this lecture he also introduces the point of view of Hilbert schemes.

And then you can read chapter $VI$, section $VI.2.2$ of "The geometry of schemes" by D. Eisenbud and J. Harris (GTM springer).

Edit: many thanks for your comments, determining the dimension of Chow varieties (the maximum of the dimension of its components) is not easy, for curves you have a paper by D. Eisenbud and J. Harris "The dimension of the Chow variety of curves." Compositio Math. 83 (1992), no. 3, 291–310. and they set a conjecture for higher dimensional subvarieties of projective spaces. I am sorry but do not know what is the state of the art.

Concerning Chow varieties vs Hilbert schemes in this paper they make the following remark: ".... The second reason for working with Chow rather than Hilbert is that the Hilbert scheme will have aditionnal components whose general are not of pure dimension (e.g. consist of the union of a reduced curve and a zero dimensional subscheme). For all we know these components may have larger dimension than those corresponding to components of the Chow variety."