  Under an explicit positivity condition, we show the first secant variety of a
linearly normal smooth variety is projectively normal, give results on the
regularity of the ideal of the secant variety, and give conditions on the
variety that are equivalent to the secant variety being arithmetically
Cohen-Macaulay. Under this same condition, we then show that if $X$ satisfies
$N_{p+2\dim(X)}$, then the secant variety satisfies $N_{3,p}$.
