On k-Jet Ampleness

Abstract

Let X be an n-dimensional projective manifold mapped into a projective space Ψ:X → ℙ_ℂ. Let L be the pullback, Ψ*Oℙ_ℂ(1), of the hyperplane section bundle. If Ψ is an embedding, L is said to be very ample. This is an intensively studied and well-understood concept. In this chapter we study a particular notion of higher-order embedding. We say that L is k-jet ample for a nonnegative integer k if, given any r integers k ₁ , ., k _r , such that $k + 1 = \sum\nolimits_{i = 1}^r {{k_i}}$ and any r distinct points {x ₁ ,. . ., x _r } ⊂ X, the evaluation map

$$ X \times \Gamma (L) \to L/L \otimes m_{{x_1}}^{{k_1}} \otimes ... \otimes m_{xr}^{{k_r}} \to 0 $$

is surjective, where m _xi . denotes the maximal ideal at x _t . Note that L is spanned (respectively, very ample) if and only if L is 0-jet ample (respectively, 1-jet ample).