Automorphism groups of semidirect products

This paper shows that if is a semidirect product of finite groups, then if and only if and for all . As an application, we investigate the automorphism group of a split metacyclic p-group for odd p. The second author is supported by the Natural Science Foundation of China (10671058).     

Linear fractional mappings: invariant sets, semigroups and commutativity

We study commutativity and embeddability (into continuous semi-groups) properties of linear fractional self-mappings of the open unit disk in the complex plane. The common thread in our approach is the classical notion of the Kœnigs function which we use in each of the three possible cases (dilation, hyperbolic and parabolic). Since we are interested in a classical subject, the paper is written in the style of a survey, in order to make it accessible to a wider audience. Therefore it contains, in addition to our new results, an exposition of most relevant facts. Dedicated to Professor Felix E. Browder with admiration and respect     

Weights of mixed tilting sheaves and geometric Ringel duality

We describe several general methods for calculating weights of mixed tilting sheaves. We introduce a notion called “non-cancellation property” which implies a strong uniqueness of mixed tilting sheaves and enables one to calculate their weights effectively. When we have a certain Radon transform, we prove a geometric analogue of Ringel duality which sends tilting objects to projective objects. We apply these methods to (partial) flag varieties and affine (partial) flag varieties and show that the weight polynomials of mixed tilting sheaves on flag and affine flag varieties are essentially given by Kazhdan-Lusztig polynomials. This verifies a mixed geometric analogue of a conjecture by W. Soergel in [10].      

Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. I

In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers and the relative Jacobian of a family of curves . As one of the applications, we show that in the case of a single curve C this action induces a -form of a Lefschetz sl₂- action on the Chow groups of C ^[N]. Another application gives a new grading on the ring CH ₀(J) of 0-cycles on the Jacobian J of C (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes in CH and in CH and prove for them analogs of the properties established in the case of the Jacobian of a single curve by Beauville in [5]. We show that our algebras of operators preserve the subrings of tautological cycles and act on them as some explicit differential operators. This work was partially supported by the NSF grant DMS-0601034.     

The base components of the dualizing sheaf of a curve on a surface

This note studies the structure of the divisorial fixed part of for a 1-connected curve D on a smooth surface S. It is shown that if the divisorial fixed part F of is non empty then it has arithmetic genus ≤ 0 and each component of F is a smooth rational curve. The structure of curves D, with non empty divisorial fixed part F for , is also described. Received: 16 August 2007     

Group Algebras: Normal Subgroups and Ideals

The standard correspondence between the normal subgroups of the group G and some ideals of the group algebra FG is described. There is the problem of what we can say (or even prove) about a two-sided ideal of that does not contain any element of the form 1 − g ≠ 0, g ∈G of the standard basis of the augmentation ideal of . The main part of the argument of [2] yields the insight that, for such an ideal I there exists an expansion such that the ideal J of spanned by I contains an element 1 − h, h ∈ H \ G. Using the ideas of [2], we construct -thick groups H such that for every ideal J ≠ (0) of there are elements 1 − h ≠ 0 in J. This construction allows many variations. Examples of simple -thick groups were pointed out in [2]. A natural class of (in general non-simple) -full groups are the normal sections of the groups

On the socles of fully invariant subgroups of Abelian p-groups

The classification of the fully invariant subgroups of a reduced Abelian p-group is a difficult long-standing problem when one moves outside of the class of fully transitive groups. In this work we restrict attention to the socles of fully invariant subgroups and introduce a new class of groups which we term socle-regular groups; this class is shown to be large and strictly contains the class of fully transitive groups. The basic properties of such groups are investigated but it is shown that the classification of even this simplified class of groups, seems extremely difficult. Received: 4 September 2008     

The automorphism group of a nonsplit metacyclic p-group

This note considers a finite group G = HK, which is a product of a subgroup H and a normal subgroup K, and determines subgroups of Aut G. The special case when G is a nonsplit metacyclic p-group, where p is odd, is then considered and the structure of its automorphism group Aut G is given. Received: 13 September 2007, Revised: 22 November 2007     

Trigonal reducible curves

Here we study the canonical model of a reducible trigonal Gorenstein curve X. We prove that the canonical model is arithmetically Cohen — Macaulay and lies in a minimal degree Hirzebruch surface, generalizing the classical theory of Maroni on smooth trigonal curves.     

Remarks on normal generation of line bundles on algebraic curves

We determine necessary and sufficient conditions for nonspecial line bundles of degree 2% - 4 and 2g - 5 being not normally generated. Furthermore, we also determine necessary and suffcient conditions for speciality 1 line bundles of degree 2g -7,2% - 8, and 2g - 9 being not normally generated.     

Linear series computing the Clifford index of a projective curve

For a (smooth irreducible) curveC of genus g and Clifford indexc>2 with a linear seriesg _d ^r computing c (so ) it is well known thatc + 2 ≤d ≤2 (c + 2), and if then 2c + 1 ≤g ≤ 2c + 4 unlessd = 2c + 4 in which caseg = 2c + 5. Let c ≥ 0 andg be integers. If 2c + 1 ≤g ≤2c + 4 we prove that for any integerd <g such thatd ≡c mod 2 andc + 2 ≤d < 2(c + 2) there exists a curve of genus g and Clifford index c with a g_d ^r computing c. Ford ≥c + 6 (i.e.r ≥ 3) we construct this curve on a surface of degree 2r-2 in ℙ^r, and ford ≥c + 8 (i.e.r ≥ 4) we show that such a curve cannot be found on a surface in ℙ^r of smaller degree. In fact, if g_d ^r computes the Clifford index c of C such thatc + 8 ≤d ≤ 2c + 3 then the birational morphism defined by this series cannot map C onto a (maybe, singular) curve contained in a surface of degree at most 2r-3 in ℙ^r.     

On the quantum homology algebra of toric Fano manifolds

We study certain algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of this algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily verifiable sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non-semisimple quantum homology, and others in which the Calabi quasi-morphism is not unique.      

Gorenstein quivers

In this paper we characterize when the path ring associated to a quiver is Gorenstein (in the sense of Iwanaga [9]). Then, by using the notion of a Gorenstein category (cf. [2]), we extend the classes of quivers whose corresponding category of representations has finite Gorenstein global dimension. This extension includes non-noetherian quivers. E. E., S.E., and J.R.G.R., partially supported by the DGI MTM2005-03227. Estrada’s work was supported by a MEC/Fulbright grant from the Spanish Secretaría de Estado de Universidades e Investigación del Ministerio de Educación y Ciencia. Received: 28 February 2006     

Uniqueness Results for the Riemann-Hilbert Problem with a Vanishing Coefficient

We study the Riemann-Hilbert problem of finding φ, ψ ∈ H^p such that their nontangential boundary values satisfy the equation

Fano 3-folds with divisible anticanonical class

We show the nonvanishing of H ⁰(X,−K _X ) for any a Fano 3-fold X for which −K _X is a multiple of another Weil divisor in Cl(X). The main case we study is Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X)=1, -factorial terminal singularities and −K _X  = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised varieties (X,A) and deduce both the nonvanishing of H ⁰(X,−K _X ) and the sharp bound (−K _X )³≥ 8/165. We find the families that can be realised in codimension up to 4.     

A note on families of hyperelliptic curves

We give a stack-theoretic proof for some results on families of hyperelliptic curves. Received: 5 February 2008     

On <Emphasis Type="Italic">pq</Emphasis>-hyperelliptic Klein surfaces

In this paper we study compact Klein surfaces of algebraic genus d > 1 admitting p- and q-hyperelliptic involutions by which we mean involutions with the orbit spaces having algebraic genera p and q. We give necessary and sufficient conditions for p, q and d to exist such surfaces. It turns out that these conditions are also sufficient for the existence of such surfaces with commuting involutions what allow us to study this class also. We study the spectrum of hyperellipticity degrees of the product of these involutions and topological type of these surfaces. G. Gromadzki was supported by the grant SAB 2005-0049 of the Spanish Ministry of Education and Sciences. E. Tyszkowska was supported by BW 5100-5-0198-6.     

Homogeneous spaces and degree 4 del Pezzo surfaces

It is known that, given a genus 2 curve , where f(x) is quintic and defined over a field K, of characteristic different from 2, and given a homogeneous space for complete 2-descent on the Jacobian of , there is a V _δ (which we shall describe), which is a degree 4 del Pezzo surface defined over K, such that . We shall prove that every degree 4 del Pezzo surface V, defined over K, arises in this way; furthermore, we shall show explicitly how, given V, to find and δ such that V =  V _δ , up to a linear change in variable defined over K. We shall also apply this relationship to Hürlimann’s example of a degree 4 del Pezzo surface violating the Hasse principle, and derive an explicit parametrised infinite family of genus 2 curves, defined over , whose Jacobians have nontrivial members of the Shafarevich-Tate group. This example will differ from previous examples in the literature by having only two -rational Weierstrass points. The author thanks EPSRC for support: grant number EP/F060661/1.     

Graph homology of the moduli space of pointed real curves of genus zero

The moduli space parameterizes the isomorphism classes of S-pointed stable real curves of genus zero which are invariant under relabeling by the involution σ. This moduli space is stratified according to the degeneration types of σ-invariant curves. The degeneration types of σ-invariant curves are encoded by their dual trees with additional decorations. We construct a combinatorial graph complex generated by the fundamental classes of strata of . We show that the homology of is isomorphic to the homology of our graph complex. We also give a presentation of the fundamental group of .