Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces

Let Pn be the n-dimensional projective space over some algebraically closed field k of characteristic 0. For an integer t ≥ 3 consider the invertible sheaf O(t) on Pn (Serre twist of the structure sheaf). Let , the dimension of the space of global sections of O(t), and let k be an integer satisfying 0 < k ≤ N − (2n + 2). Let P1,…,Pk be general points on Pn and let π : X → Pn be the blowing-up of Pn at those points. Let Ei = π−1(Pi) with 1 ≤ i ≤ k be the exceptional divisor. Then M = π*(O(t)) ⊗ OX(−E1 — … — Ek) is a very ample invertible sheaf on X.

The family of lines on the Fano threefold V5

A smooth projective algebraic 3-fold V over the field C is called a Fano 3-fold if the anticanonical divisor — Kv is ample. The integer g = g(V) = ½(- Kv)3 is called the genus of the Fano 3-fold V. The maximal integer r ≧ 1 such that ϑ(— Kv)≃ ℋ r for some (ample) invertible sheaf ℋ ε Pic V is called the index of the Fano 3-fold V. Let V be a Fano 3-fold of the index r = 2 and the genus g = 21 which has the second Betti number b2(V) = 1. Then V can be embedded in P6 with degree 5, by the linear system |ℋ|, where ϑ(— Kv)≃ ℋ2 (see Iskovskih [5]). We denote this Fano 3-fold V by V5.