Positivity of Chern Classes for Reflexive Sheaves on P^N

It is well known that the Chern classes $c_i$ of a rank $n$ vector bundle on \PP^N, generated by global sections, are non-negative if $i\leq n$ and vanish otherwise. This paper deals with the following question: does the above result hold for the wider class of reflexive sheaves? We show that the Chern numbers $c_i$ with $i\geq 4$ can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for $i\leq 3$ we show positivity of the $c_i$ with weaker hypothesis. We obtain lower bounds for $c_1$, $c_2$ and $c_3$ for every reflexive sheaf \FF which is generated by H^0\FF on some non-empty open subset and completely classify sheaves for which either of them reach the minimum allowed, or some value close to it.Comment: 16 pages, no figure

Hodge metrics and positivity of direct images

Building on Fujita-Griffiths method of computing metrics on Hodge bundles, we show that the direct image of an adjoint semi-ample line bundle by a projective submersion has a continuous metric with Griffiths semi-positive curvature. This shows that for every holomorphic semi-ample vector bundle $E$ on a complex manifold, and every positive integer $k$, the vector bundle $S^kE\otimes\det E$ has a continuous metric with Griffiths semi-positive curvature. If $E$ is ample on a projective manifold, the metric can be made smooth and Griffiths positive.Comment: revised and expanded version of "A positivity property of ample vector bundles

On the intersection of the curves through a set of points in P^2

Given a set of points in P^2, we consider the common zeros of the set of curves of a given degree passing through those points. For general sets of points, these zero sets have the expected dimension and are smooth. In fact, given graded Betti numbers, for any arrangement of points whose ideal has those graded Betti numbers, general among such arrangements, the zero sets have the expected dimension and are smooth.Comment: minor changes; final version, to appear in JPA

A simple remark on a flat projective morphism with a Calabi-Yau fiber

If a K3 surface is a fiber of a flat projective morphisms over a connected noetherian scheme over the complex number field, then any smooth connected fiber is also a K3 surface. Observing this, Professor Nam-Hoon Lee asked if the same is true for higher dimensional Calabi-Yau fibers. We shall give an explicit negative answer to his question as well as a proof of his initial observation.Comment: 8 pages, main theorem is generalized, one more remark is added, mis-calculation and typos are corrected etc

Remarks on hard Lefschetz conjectures on Chow groups

We propose two conjectures of Hard Lefschetz type on Chow groups and prove them for some special cases. For abelian varieties, we shall show they are equivalent to well-known conjectures of Beauville and Murre.Comment: to appear in Sciences in China, Ser. A Mathematic

Immature oocytes grow during in vitro maturation culture

BACKGROUND. Oocyte competence for maturation and embryogenesis is associated with oocyte diameter in many mammals. This study aimed to test whether such a relationship exists in humans and to quantify its impact upon in vitro maturation (IVM). METHODS. We used computer-assisted image analysis daily to measure average diameter, zona thickness and other parameters in oocytes. Immature oocytes originated from unstimulated patients with polycystic ovaries, and from stimulated patients undergoing ICSI. They were cultured with or without meiosis activating sterol (FF-MAS). Oocytes maturing in vitro were inseminated using ICSI and embryo development was monitored. A sample of freshly collected in vivo matured oocytes from ICSI patients were also measured. RESULTS. Immature oocytes were usually smaller at collection than in vivo matured oocytes. Capacity for maturation was related to oocyte diameter and many oocytes grew in culture. FF-MAS stimulated growth in ICSI derived oocytes, but only stimulated growth in PCO derived oocytes if they eventually matured in vitro. Oocytes degenerating showed cytoplasmic shrinkage. Neither zona thickness, perivitelline space, nor the total diameter of the oocyte including the zona were informative regarding oocyte maturation capacity. CONCLUSIONS. Immature oocytes continue growing during maturation culture. FF-MAS promotes oocyte growth in vitro. Oocytes from different sources have different growth profiles in vitro. Measuring diameters of oocytes used in clinical IVM may provide additional non-invasive information that could potentially identify and avoid the use of oocytes that remain in the growth phase

On special quadratic birational transformations of a projective space into a hypersurface

We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by quadratic forms by showing that there are only four examples having general hyperplane sections of Severi varieties as base loci.Comment: Accepted for publication in Rendiconti del Circolo Matematico di Palerm

A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.Comment: Published in the proceedings of 2017 Southern Regional Algebra Conferenc

Relative Riemann-Zariski spaces

In this paper we study relative Riemann-Zariski spaces attached to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition theorem which asserts that any separated morphism between quasi-compact and quasi-separated schemes factors as a composition of an affine morphism and a proper morphism. (In particular, we obtain a new proof of Nagata's compactification theorem.)Comment: 30 pages, the final version, to appear in Israel J. of Mat

Ground State Entropy of Potts Antiferromagnets on Cyclic Polygon Chain Graphs

We present exact calculations of chromatic polynomials for families of cyclic graphs consisting of linked polygons, where the polygons may be adjacent or separated by a given number of bonds. From these we calculate the (exponential of the) ground state entropy, $W$, for the q-state Potts model on these graphs in the limit of infinitely many vertices. A number of properties are proved concerning the continuous locus, ${\cal B}$, of nonanalyticities in $W$. Our results provide further evidence for a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where $S_0 > 0$.Comment: 27 pages, Latex, 17 figs. J. Phys. A, in pres