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The purpose of this paper is to prove that one can read off the gonality sequence of a smooth projective curve from syzygies of secant varieties of the curve embedded by a line bundle of sufficiently large degree. More precisely, together with Ein-Niu-Park's theorem, our main result shows that the gonality sequence of a smooth projective curve completely determines the shape of the minimal free resolutions of secant varieties of the curve of sufficiently large degree. This is a natural generalization of the gonality conjecture on syzygies of smooth projective curves established by Ein-Lazarsfeld and Rathmann to the secant varieties., 22 pages, any comments are welcome

We give simple geometric proofs of Aprodu-Farkas-Papadima-Raicu-Weyman's theorem on syzygies of tangent developable surfaces of rational normal curves and Raicu-Sam's result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green's conjecture for general curves of genus $g$ over an algebraically closed field $\mathbf{k}$ with $\operatorname{char}(\mathbf{k}) = 0$ or $\operatorname{char}(\mathbf{k}) \geq \lfloor (g-1)/2 \rfloor$. We also show the arithmetic normality of tangent developable surfaces of arbitrary smooth projective curves of large degree., 16 pages. Comments are welcome

Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension $n$ with $H$ fundamental divisor. Assume $h^0(X, H) \geq n -2$. We prove that a general element of the linear system $|mH|$ has at worst canonical singularities for any integer $m \geq 1$. When $X$ has terminal singularities and $n \geq 5$, we show that a general element of $|mH|$ has at worst terminal singularities for any integer $m \geq 1$. When $n=4$, we give an example of Gorenstein terminal Fano fourfold $X$ such that a general element of $|H|$ does not have terminal singularities., 11 pages, contributed to Moscow-Shanghai-Pohang conference proceedings

The purpose of this paper is to establish a subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As applications, we obtain a product formula of the restricted canonical volumes for algebraic fiber spaces and a sufficient condition for an algebraic fiber space to be birationally isotrivial in terms of Okounkov bodies when a general fiber is of general type. Furthermore, we also prove the subadditivity of the numerical Iitaka dimensions for algebraic fiber spaces, and this confirms some numerical variants of the Iitaka conjecture. We hope that our results would provide a new approach toward the Iitaka conjecture.

The purpose of this paper is to establish a Castelnuovo-Mumford regularity bound for threefolds with mild singularities. Let $X$ be a non-degenerate normal projective threefold in $\mathbb{P}^r$ of degree $d$ and codimension $e$. We prove that if $X$ has rational singularities, then $\text{reg}(X) \leq d-e+2$. Our bound is very close to a sharp bound conjectured by Eisenbud-Goto. When $e=2$ and $X$ has Cohen-Macaulay Du Bois singularities, we obtain the conjectured bound $\text{reg}(X) \leq d-1$, and we also classify the extremal cases. To achieve these results, we bound the regularity of fibers of a generic projection of $X$ by using Loewy length, and also bound the dimension of the varieties swept out by secant lines through the singular locus of $X$., Comment: 17 pages, to appear in Adv. Math

To seek for the useful numerical analogues to the Iitaka dimension, various numerical Iitaka dimensions have been defined from a number of different perspectives. It has been accepted that all the known numerical Iitaka dimensions coincide with each other until the recent discovery of a counterexample constructed by Lesieutre. In this paper, we prove that many of them still coincide with the numerical Iitaka dimension introduced by Boucksom-Demailly-P\u{a}un-Peternell. On the other hand, we show that some other numerical Iitaka dimensions introduced by Nakayama and Lehmann can be arbitrarily larger than the rest of numerical Iitaka dimensions. We also study some properties of abundant divisors., Comment: 22 pages, comments welcome

The aim of this paper is to study the Okounkov bodies associated to abundant divisors. As a main result, we prove that the valuative Okounkov bodies of an abundant divisor encode all the numerical properties. We apply this result to recover the asymptotic base loci of an abundant divisor from the valuative Okounkov bodies. We also give a criterion of when the valuative and limiting Okounkov bodies of an abundant divisor coincide by comparing their Euclidean volumes. To obtain these results, we prove some variants of Fujita's approximations for Okounkov bodies using Iitaka fibrations., Comment: 28 pages, comments welcome, we have clarified the issue on numerical Iitaka dimensions

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., 10 pages. to appear in J. Algebra

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., 16 pages, Changed Section 5 and corrected typos, to appear in Bull. Korean. Math. Soc. (special volume for Magadan Conference)

An Okounkov body is a convex subset in Euclidean space associated to a big divisor on a smooth projective variety with respect to an admissible flag. In this paper, we introduce two convex bodies associated to pseudoeffective divisors, called the valuative Okounkov bodies and the limiting Okounkov bodies, and show that these convex bodies reflect the asymptotic properties of pseudoeffective divisors as in the case with big divisors. Our results extend the works of Lazarsfeld-Musta\c{t}\u{a} and Kaveh-Khovanskii. For this purpose, we define and study special subvarieties, called the Nakayama subvarieties and the positive volume subvarieties, associated to pseudoeffective divisors., Comment: 24 pages. Final version. To appear in J. London Math. Soc

We prove that the redundant blow-up preserves the finite generation of the Cox ring of a rational surface under a suitable assumption, and we study the birational structure of Mori dream rational surfaces via redundant blow-ups. It turns out that the redundant blow-up completely characterizes birational morphisms of Mori dream rational surfaces with anticanonical Iitaka dimension $0$. As an application, we construct new Mori dream rational surfaces with anticanonical Iitaka dimension $0$ and $-\infty$ of arbitrarily large Picard number., 16 pages. We have divided it into two parts and changed the title. This part will appear in Trans. Amer. Math. Soc