O_X regularity bound for smooth varieties with classification of extremal and next to extremal examples

For a smooth variety $X$ and a very ample line bundle $\mathcal L$, $\mathcal O_X$ is $m$-regular with respect to $\mathcal L$ if $H^i(X, \mathcal L^{m-i})=0$ for all $i\ge 1$ and $reg_{\mathcal L}(\mathcal O_X):=\{m \mid \mathcal O_X \text{is $m$-regular}\}.$ For an embedded $n$-dimensional smooth variety $X$ in $\Bbb P^{n+e}$, Eisenbud-Goto conjecture tells us that $reg_{\mathcal L}(\mathcal O_X)\le deg(X)-codim(X)$. We will show that this conjecture is true for smooth varieties and classify boundary cases. On the other hand, there are many counterexamples for singular varieties due to McCullough and Peeva so that it would be desirable to understand the dichotomy between singular cases and smooth cases and how hard to show Castelnuovo normality bound for smooth varieties.Non UBCUnreviewedAuthor affiliation: KAISTFacult