First properties

Surfaces of degreedin ℙd

Recall that a subvariety X ⊂ ℙn is called nondegenerate if it is not contained in a proper linear subspace. All varieties we consider here are assumed to be reduced. Let d = deg(X). We have the following well-known (i.e. can be found in modern text-books, e.g. [264], [275]) result.

Theorem 8.1.1

Let X be an irreducible nondegenerate subvariety of ℙn of dimension k and degree d. Then d ≥ n − k + 1, and the equality holds only in one of the following cases:

1. (i) X is a quadric hypersurface;

2. (ii) X is a Veronese surface in ℙ5;

3. (iii) X is a cone over a Veronese surface in ℙ5;

4. (iv) X is a rational normal scroll.

Recall that a rational normal scroll is defined as follows. Choose k disjoint linear subspaces L1, …, Lk in ℙn which together span the space. Let ai = dim Li. We have. Consider Veronese maps vai: ℙ1 → Li and define, …, to be the union of linear subspaces spanned by the points va1 (x), …, vak (x), where x ∈ ℙ1. It is clear that dim, …, = k and it is easy to see that deg, …, ak;n = a1 + … + ak and dim, …, ak;n = k. In this notation, it is assumed that a1 ≤ a2 ≤ … ≤ ak.