Consider globally complete intersections in $\mathbb{P}^n$, of codimension $k$, of some fixed multi-degree $(d\_1,\dots,d\_k)$. 


1. Is there some nice (i.e. "explicit") parameter space for them?
(even if all the $d\_i$'s are distinct one cannot take just the product of linear systems :)
Or, does the corresponding locus in the Hilbert scheme admit some nice/explicit description?
(a [relevant question](https://mathoverflow.net/questions/17110/complete-intersections-and-flat-families) on this locus in Hilbert scheme)
2. Consider the discriminant in the parameter space of complete intersections, i.e. the locus parametrizing singular c.i.'s. I need standard results: it is an irreducible, reduced hypersurface, singular in codimension one etc. (I guess many of these properties do not depend on the particular choice of parameter space.) Where can I read about this? 
Somehow I did not find this in Gel'fand-Kapranov-Zelevinsky. And the book of Looijenga treats the local case only.


a somewhat related [question](https://mathoverflow.net/questions/97865/what-can-be-reached-by-flat-degeneration-of-globally-complete-intersection)



Parameter space for complete intersections and their discriminant
The description of the Hilbert scheme of complete intersections (obtained by taking in an iterative way open subsets of grassmannian bundles, as explained in the answer above) may be found in part 2.2 of my thesis <http://www.math.ens.fr/~obenoist/articles/Thesefinale.pdf>. If the distinct degrees are $\delta\_1<\dots<\delta\_r$, at the $i$-th step you construct a grassmannian bundle corresponding to choosing the equations of degree $\delta\_i$ up to multiples of the equations of lower degree, and you remove the closed locus where your equations do not define a complete intersection.


Note that this space is smooth. It is not complete unless $k=1$. However, the construction above provides an explicit compactification if $d\_1\leq d\_2=\dots=d\_k$. This compactification is studied in part 2 of arXiv:1111.1589.


The Picard group of this Hilbert scheme is easy to describe (its rank increases by one with each grassmannian bundle and it is not changed when removing the small closed loci).The class of the discriminant in this Picard group (i.e. the degrees of the discriminant) is calculated in arXiv:1009.0704 Théorème 1.3, building on Gel'fand, Kapranov and Zelevinsky's work. 


