arXiv:math/0701409v2 [math.AG] 10 Sep 2007 On the Alexander-Hirschowitz Theorem Maria Chiara Brambilla and Giorgio Ottaviani Abstract The Alexander-Hirschowitz theorem says that a general collection of k double points in P n imposes independent conditions on homogeneous polynomials of de- gree d with a well known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a self-contained proof based mainly on previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case d = 3, where our proof is shorter. We end with an account of the history of the work on this problem. AMS Subject Classification: 01-02, 14C20, 15A72, 14M17 Both authors are partially supported by Italian MUR and are members of GN- SAGA. 1 Introduction The aim of this paper is to expose a proof of the following theorem. Theorem 1.1 (Alexander-Hirschowitz) Let X be a general collection of k double points in P n = P(V ) (over an algebraically closed field of characteristic zero) and let S d V ∨ be the space of homogeneous polynomials of degree d. Let I X (d) ⊆ S d V ∨ be the subspace of polynomials through X , that is with all first partial derivatives vanishing at the points of X . Then the subspace I X (d) has the expected codimension min  (n + 1)k, ( n+d n )  except in the following cases • d =2, 2 ≤ k ≤ n; • n =2,d =4,k = 5; • n =3,d =4,k = 9; • n =4,d =3,k = 7; • n =4,d =4,k = 14. We remark that the case n = 1 is the only one where the assumption that X is general is not necessary. More information on the exceptional cases is contained in Section 3. This theorem has an equivalent formulation in terms of higher secant varieties. Given a projective variety Y , the k-secant variety σ k (Y ) is the Zariski closure of the union of all the linear spans <p 1 ,...,p k > where p i ∈ Y (see [Ru] or [Z]). In particular σ 1 (Y ) coincides with Y and σ 2 (Y ) is the usual secant variety. Consider the Veronese embedding V d,n ⊂ P m of degree d of P n , that is the image of the linear system given by all homogeneous polynomials of degree d, where m = ( n+d n ) − 1. It is easy to check that dim σ k (V d,n ) ≤ min ((n + 1)k − 1,m) and when the equality holds we say that σ k (V d,n ) has the expected dimension. 1