Regularity and normality of the secant variety to a projective curve

For a smooth curve of genus g embedded by a line bundle of degree at least 2g+3 we show that the ideal sheaf of the secant variety is 5-regular. This bound is sharp with respect to both the degree of the embedding and the bound on the regularity. Further, we show that the secant variety is projectively normal for the generic embedding of degree at least 2g+3.