Projective surfaces with K-very ample line bundles of degree ≤ 4K + 4

Abstract

A line bundle, L, on a smooth, connected projective surface, S, is defined [7] to be k-very ample for a non-negative integer, k, if given any 0-dimensional sub-scheme with length , it follows that the restriction map is onto. L is 1-very ample (respectively 0-very ample) if and only if L is very ample (respectively spanned at all points by global sections). For a smooth surface, S, embedded in projective space by | L | where L is very ample, L being k-very ample is equivalent to there being no k-secant Pk−1 to S containing ≥ k + 1 points of S.