Very ample line bundles on regular surfaces obtained by projection

AbstractWe work over an algebraically closed field of arbitrary characteristic. Let X⊆PN be a smooth projective surface with very ample line bundle L:=OX(1), of degree d and sectional genus g. Consider the blowing-up σ:Xˆ→X at distinct points x1,…,xm∈X with the exceptional divisors E1,…,Em and let Lˆ be the line bundle σ∗L⊗OXˆ(−E1−⋯−Em) on Xˆ. The purpose here is to give a necessary and sufficient condition for Lˆ to be very ample in terms of the configuration of x1,…,xm, for surfaces with h1(X,OX)=0 and m⩽d−2g−1. The key tool for the proof is the linear projection from a point of X. As an application, we will determine some surfaces of sectional genus 2 or 3