A bound for Castelnuovo-Mumford regularity by double point divisors

Let $X \subseteq \mathbb{P}^r$ 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 $\text{reg} (\mathcal{O}_X) \leq d-e$, 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 $\text{reg} (\mathcal{O}_X)$ 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 establish that $\text{reg}(X) \leq n(d-2)+1$, 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 $\text{reg}(X) \leq d-1+m$, where $m$ is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projection, we also obtain a slightly better bound for $\text{reg}(X)$ under suitable assumptions.Comment: 23 pages. This paper has been largely rewritten after McCullough-Peeva's counterexamples to the Eisenbud-Goto regularity conjecture, which appeared in J. Amer. Math. Soc. in 2018. We also added new results on the regularity of smooth projective varieties of arbitrary dimensio

Hilbert functions of Cox rings of del Pezzo surfaces

To study syzygies of the Cox rings of del Pezzo surfaces, we calculate important syzygetic invariants such as the Hilbert functions, the Green-Lazarsfeld indices, the projective dimensions, and the Castelnuovo-Mumford regularities. Using these computations as well as the natural multigrading structures by the Picard groups of del Pezzo surfaces and Weyl group actions on Picard lattices, we determine the Betti diagrams of the Cox rings of del Pezzo surfaces of degree at most four.Comment: 15 pages, final versio

Characterization of log del Pezzo pairs via anticanonical models

There are several variations of the definition of log del Pezzo pairs in the literature. We define their suitable smooth models, and we show that they are the same. In particular, we obtain a characterization of smooth log del Pezzo pairs in terms of anticanonical models. As applications, we classify non-rational weak log canonical del Pezzo pairs, and we prove that every surface of globally F-regular type is of Fano type.Comment: 18 pages. Comments are welcome. Lemma 7.5 is correcte

Geometric properties of projective manifolds of small degree

The aim of this paper is to study geometric properties of non-degenerate smooth projective varieties of small degree from a birational point of view. First, using the positivity property of double point divisors and the adjunction mappings, we classify smooth projective varieties in $\mathbb P^r$ of degree $d \leq r+2$, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalization of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in $\mathbb P^r$ of degree $d \leq r$ with counterexamples for $d=r+1, r+2$. On the other hand, we prove that a non-uniruled smooth projective variety in $\mathbb P^r$ of dimension $n$ and degree $d \leq n(r-n)+2$ is Calabi-Yau, and give an example that shows this bound is also sharp.Comment: To appear in Math. Proc. Cambridge Philos. So

Classification and syzygies of smooth projective varieties with 2-regular structure sheaf

The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo-Mumford regularity.Comment: 13 page

A Castelnuovo-Mumford regularity bound for scrolls

Let $X \subseteq \mathbb{P}^r$ be a scroll of codimension $e$ and degree $d$ over a smooth projective curve of genus $g$. The purpose of this paper is to prove a linear Castelnuovo-Mumford regularity bound that reg$(X) \leq d-e+1+g(e-1)$. This bound works over an algebraically closed field of arbitrary characteristic.Comment: 10 pages. to appear in J. Algebr

Potentially non-klt locus and its applications

We introduce the notion of potentially klt pairs for normal projective varieties with pseudoeffective anticanonical divisor. The potentially non-klt locus is a subset of $X$ which is birationally transformed precisely into the non-klt locus on a $-K_X$-minimal model of $X$. We prove basic properties of potentially non-klt locus in comparison with those of classical non-klt locus. As applications, we give a new characterization of varieties of Fano type, and we also improve results on the rational connectedness of uniruled varieties with pseudoeffective anticanonical divisor.Comment: 25 pages. We slightly modified the definitions of potentially klt pairs and potentially non-klt loci, and corrected a gap in the proof of Proposition 4.

Local numerical equivalences and Okounkov bodies in higher dimensions

We continue to explore the numerical nature of the Okounkov bodies focusing on the local behaviors near given points. More precisely, we show that the set of Okounkov bodies of a pseudoeffective divisor with respect to admissible flags centered at a fixed point determines the local numerical equivalence class of divisors which is defined in terms of refined divisorial Zariski decompositions. Our results extend Ro\'{e}'s work on surfaces to higher dimensional varieties although our proof is essentially different in nature.Comment: 20 page

Singularities and syzygies of secant varieties of nonsingular projective curves

In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus $g$ is greater than $2g+2k+p$ for nonnegative integers $k$ and $p$, then the $k$-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen--Macaulay, and satisfies the property $N_{k+2, p}$. In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo--Mumford regularities are also obtained as well. As one of the main technical ingredients, we establish a vanishing theorem on the Cartesian products of the curve, which may have independent interests and may find applications elsewhere.Comment: 36 pages. Comments are welcom

Okounkov bodies and Zariski decompositions on surfaces

The purpose of this paper is to investigate the close relation between Okounkov bodies and Zariski decompositions of pseudoeffective divisors on smooth projective surfaces. Firstly, we completely determine the limiting Okounkov bodies on such surfaces, and give applications to Nakayama constants and Seshadri constants. Secondly, we study how the shapes of Okounkov bodies change as we vary the divisors in the big cone.Comment: 16 pages, Changed Section 5 and corrected typos, to appear in Bull. Korean. Math. Soc. (special volume for Magadan Conference