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Let $X \subseteq \mathbb{P}^r$ be a non-degenerate smooth projective varietyof dimension $n$, codimension $e$, and degree $d$ defined over an algebraicallyclosed field of characteristic zero. In this paper, we first show that$\text{reg} (\mathcal{O}_X) \leq d-e$, and classify the extremal and the nextto extremal cases. Our result reduces the Eisenbud-Goto regularity conjecturefor the smooth case to the problem finding a Castelnuovo-type bound fornormality. It is worth noting that McCullough-Peeva recently constructedcounterexamples to the regularity conjecture by showing that $\text{reg}(\mathcal{O}_X)$ 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 establishthat $\text{reg}(X) \leq n(d-2)+1$, which improves previous results obtained byMumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford'smethod on double point divisors, we prove that $\text{reg}(X) \leq d-1+m$,where $m$ is an invariant arising from double point divisors associated toouter general projections. Using double point divisors associated to innerprojection, we also obtain a slightly better bound for $\text{reg}(X)$ undersuitable assumptions.

A bound for Castelnuovo-Mumford regularity by double point divisors