General Blowings-up of ℙ² and Injectivity of Gauss Maps

Abstract

We consider the blowing-up Y _k of the projective plane along k general points P ₁,...,P _k . Let π _k : Y _k →$\mathbb{P}$ ² be the projection map and E _i the exceptional divisor corresponding to P _i for 1≤i≤k. For m≥2 and k≤m(m+3)/2−4 let $\mathcal{M}$ _k be the invertible sheaf π _k ^*($\mathcal{O}$ $\mathbb{P}$ ²(m))⊗$\mathcal{O}$ _{Y k} (−E ₁−···−E _k ) on Y _k , and let φ_k: Y _k →$\mathbb{P}$ ^N be the morphism corresponding to $\mathcal{M}$ _k . As φ _k is a local embedding, the Gauss map γ _k corresponding to $\mathcal{M}$ _k is defined on Y _k by γ _k (x)=(d _x φ _k )(T _x (Y _k )) for all x∈Y _k . We prove that this Gauss map γ _k is injective.