Proofs

Theorem 11.3 Consider n distinct points

Proof At first, we prove that the ideal J ⊂ S</> is homogeneous, that is to say, if

A remarkable property of homogeneous ideals in polynomial rings is that they can be generated by homogeneous polynomials. Secondly, we prove that

To simplify notation, set

Now, we prove that there exists a monomial ideal

is flat, with fibers of dimension 0 and degree r + s.

If one or more points among the Qj 's belong to some lines among l1, …,lr then for some values we obtain some double points, but the family is still flat as a straightforward computation shows.

If one point among the Qj 's or one among the Pi 's is the origin, then again the family is flat for the same reasons as before.

Theorem 11.8 In the hypotheses and notation of Theorem 11.7, for every i = 1,…,r it holds

Proof The hypotheses guarantee that the polynomial ci is equal to

Proof The existence and uniqueness of F is a consequence of the isomorphism

By its properties, we have that F – Fi ε Ji, for i = 1,…, m. Then, we impose that NF(F1 + H – Fi) = 0 in R/Ji. By rewriting the polynomial F1 + H – Fi modulo Gi we get a polynomial with coefficients that are linear polynomials in the variables ar1 + 1,…, ar.