Maximal lengths of exceptional collections of line bundles

In this paper we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant $c>\frac34$ there exist infinitely many toric Fano varieties $Y$ with Picard number three, such that the maximal length of exceptional collection of line bundles on $Y$ is strictly less than c\rk K_0(Y). To obtain varieties without exceptional collections of line bundles, it suffices to put $c=1.$ On the other hand, we prove that for any toric nef-Fano DM stack $Y$ with Picard number three, there exists a strong exceptional collection of line bundles on $Y$ of length at least \frac34 \rk K_0(Y). The constant $\frac34$ is thus maximal with this property.Comment: 27 pages, no figures; misprints and typos corrected, an arithmetic mistake in the proof of Theorem 6.2 corrected, consequently Theorem 6.3 slightly modified, new Lemma 4.4 added, description of the constructed varieties extended, references adde

Development of Cu-spin correlation in Bi_1.74_Pb_0.38_Sr_1.88_Cu_1-y_Zn_y_O_6+d_ high-temperature superconductors observed by muon spin relaxation

A systematic muon-spin-relaxation study in Bi-2201 high-Tc cuprates has revealed for the first time that the Cu-spin correlation (CSC) is developed at low temperatures below 2 K in a wide range of hole concentration where superconductivity appears. The CSC tends to become weak gradually with increasing hole-concentration. Moreover, CSC has been enhanced through the 3% substitution of Zn for Cu. These results are quite similar to those observed in La-214 high-Tc cuprates. Accordingly, it has been suggested that the intimate relation between the so-called spin-charge stripe correlations and superconductivity is a universal feature in hole-doped high-Tc cuprates. Furthermore, apparent development of CSC, which is suppressed through the Zn substitution oppositely, has been observed in non-superconducting heavily overdoped samples, being argued in the context of a recently proposed ferromagnetic state in heavily overdoped cuprates.Comment: 6 pages, 5 figure

Characterization of the 4-canonical birationality of algebraic threefolds

In this article we present a 3-dimensional analogue of a well-known theorem of E. Bombieri (in 1973) which characterizes the bi-canonical birationality of surfaces of general type. Let $X$ be a projective minimal 3-fold of general type with $\mathbb{Q}$-factorial terminal singularities and the geometric genus $p_g(X)\ge 5$. We show that the 4-canonical map $\phi_4$ is {\it not} birational onto its image if and only if $X$ is birationally fibred by a family $\mathscr{C}$ of irreducible curves of geometric genus 2 with $K_X\cdot C_0=1$ where $C_0$ is a general irreducible member in $\mathscr{C}$.Comment: 25 pages, to appear in Mathematische Zeitschrif

Hole-trapping by Ni, Kondo effect and electronic phase diagram in non-superconducting Ni-substituted La2-xSrxCu1-yNiyO4

In order to investigate the electronic state in the normal state of high-Tc cuprates in a wide range of temperature and hole-concentration, specific-heat, electrical-resistivity, magnetization and muon-spin-relaxation (muSR) measurements have been performed in non-superconducting Ni-substituted La2-xSrxCu1-yNiyO4 where the superconductivity is suppressed through the partial substitution of Ni for Cu without disturbing the Cu-spin correlation in the CuO2 plane so much. In the underdoped regime, it has been found that there exist both weakly localized holes around Ni and itinerant holes at high temperatures. With decreasing temperature, all holes tend to be localized, followed by the occurrence of variable-range hopping conduction at low temperatures. Finally, in the ground state, it has been found that each Ni2+ ion traps a hole strongly and that a magnetically ordered state appears. In the overdoped regime, on the other hand, it has been found that a Kondo-like state is formed around each Ni2+ spin at low temperatures. In conclusion, the ground state of non-superconducting La2-xSrxCu1-yNiyO4 changes upon hole doping from a magnetically ordered state with the strong hole-trapping by Ni2+ to a metallic state with Kondo-like behavior due to Ni2+ spins, and the quantum phase transition is crossover-like due to the phase separation into short-range magnetically ordered and metallic regions.Comment: 9 pages, 8 figures, accepted for publication in Phys. Rev.

Three embeddings of the Klein simple group into the Cremona group of rank three

We study the action of the Klein simple group G consisting of 168 elements on two rational threefolds: the three-dimensional projective space and a smooth Fano threefold X of anticanonical degree 22 and index 1. We show that the Cremona group of rank three has at least three non-conjugate subgroups isomorphic to G. As a by-product, we prove that X admits a Kahler-Einstein metric, and we construct a smooth polarized K3 surface of degree 22 with an action of the group G.Comment: 43 page

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

Logarithmic Moduli Spaces for Surfaces of Class VII

In this paper we describe logarithmic moduli spaces of pairs (S,D) consisting of minimal surfaces S of class VII with positive second Betti number b_2 together with reduced divisors D of b_2 rational curves. The special case of Enoki surfaces has already been considered by Dloussky and Kohler. We use normal forms for the action of the fundamental group of the complement of D and for the associated holomorphic contraction germ from (C^2,0) to (C^2,0).Comment: Minor correction of the dimension of the moduli spac

Defect and Hodge numbers of hypersurfaces

We define defect for hypersurfaces with A-D-E singularities in complex projective normal Cohen-Macaulay fourfolds having some vanishing properties of Bott-type and prove formulae for Hodge numbers of big resolutions of such hypersurfaces. We compute Hodge numbers of Calabi-Yau manifolds obtained as small resolutions of cuspidal triple sextics and double octics with higher A_j singularities.Comment: 25 page