A bound for Castelnuovo-Mumford regularity by double point divisors

Let X⊆Pr be a non-degenerate smooth projective variety of dimension n, codimension e, and degree d defined over an algebraically closed field of characteristic zero. In this paper, we first show that reg(OX)≤d−e, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that reg(OX) is not even bounded above by any polynomial function of d when X is not smooth. For a normality bound in the smooth case, we establish that reg(X)≤n(d−2)+1, which improves previous results obtained by Mumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford's method on double point divisors, we prove that reg(X)≤d−1+m, where m is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projections, we also obtain a slightly better bound for reg(X) under suitable assumptions.