103 research outputs found
A bound for Castelnuovo-Mumford regularity by double point divisors
Let be a non-degenerate smooth projective variety
of dimension , codimension , and degree defined over an algebraically
closed field of characteristic zero. In this paper, we first show that
, 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 is not even bounded above by any polynomial function of
when is not smooth. For a normality bound in the smooth case, we establish
that , 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 ,
where 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 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
of degree , 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 of degree with
counterexamples for . On the other hand, we prove that a
non-uniruled smooth projective variety in of dimension and
degree 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 be a scroll of codimension and degree
over a smooth projective curve of genus . The purpose of this paper is to
prove a linear Castelnuovo-Mumford regularity bound that reg. This bound works over an algebraically closed field of arbitrary
characteristic.Comment: 10 pages. to appear in J. Algebr
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
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 which is birationally transformed precisely into the
non-klt locus on a -minimal model of . 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.
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 is greater than for nonnegative
integers and , then the -th secant variety of the curve has normal Du
Bois singularities, is arithmetically Cohen--Macaulay, and satisfies the
property . 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
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