The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

Let $$f:S'\longrightarrow S$$ f : S ′ ⟶ S be a cyclic branched covering of smooth projective surfaces over $${\mathbb {C}}$$ C whose branch locus $$\Delta \subset S$$ Δ ⊂ S is a smooth ample divisor. Pick a very ample complete linear system $$|{\mathcal {H}}|$$ | H | on S, such that the polarized surface $$(S, |{\mathcal {H}}|)$$ ( S , | H | ) is not a scroll nor has rational hyperplane sections. For the general member $$[C]\in |{\mathcal {H}}|$$ [ C ] ∈ | H | consider the $$\mu _{n}$$ μ n -equivariant isogeny decomposition of the Prym variety $${{\,\mathrm{Prym}\,}}(C'/C)$$ Prym ( C ′ / C ) of the induced covering $$f:C'{:}{=}f^{-1}(C)\longrightarrow C$$ f : C ′ : = f - 1 ( C ) ⟶ C : $$\begin{aligned} {{\,\mathrm{Prym}\,}}(C'/C)\sim \prod _{d|n,\ d\ne 1}{\mathcal {P}}_{d}(C'/C). \end{aligned}$$ Prym ( C ′ / C ) ∼ ∏ d | n , d ≠ 1 P d ( C ′ / C ) . We show that for the very general member $$[C]\in |{\mathcal {H}}|$$ [ C ] ∈ | H | the isogeny component $${\mathcal {P}}_{d}(C'/C)$$ P d ( C ′ / C ) is $$\mu _{d}$$ μ d -simple with $${{\,\mathrm{End}\,}}_{\mu _{d}}({\mathcal {P}}_{d}(C'/C))\cong {\mathbb {Z}}[\zeta _{d}]$$ End μ d ( P d ( C ′ / C ) ) ≅ Z [ ζ d ] . In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map $${\mathcal {P}}_{d}(C'/C)\subset {{\,\mathrm{Jac}\,}}(C')\longrightarrow {{\,\mathrm{Alb}\,}}(S')$$ P d ( C ′ / C ) ⊂ Jac ( C ′ ) ⟶ Alb ( S ′ ) .

Cracked polytopes and Fano toric complete intersections

We introduce the notion of cracked polytope, and – making use of joint work with Coates and Kasprzyk—construct the associated toric variety X as a subvariety of a smooth toric variety Y under certain conditions. Restricting to the case in which this subvariety is a complete intersection, we present a sufficient condition for a smoothing of X to exist inside Y. We exhibit a relative anti-canonical divisor for this smoothing of X, and show that the general member is simple normal crossings.

Plane branches with Newton non-degenerate polars

To an equisingularity class of complex plane branches, described by its multiplicity [Formula: see text] and characteristic exponents [Formula: see text], [Formula: see text], there is a naturally associated family [Formula: see text] of equations containing a complete set of analytic representatives for all branches of the class. We show in this paper that the general polar curve of any member of [Formula: see text] is Newton degenerate, except when [Formula: see text], in which case the general member of [Formula: see text] corresponds to a curve which has a Newton non-degenerate general polar curve with a fixed Newton polygon, or when [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], with [Formula: see text] and [Formula: see text] is odd, in which case [Formula: see text] has a subset containing a complete set of analytic representatives for all branches of the class whose general member has also a Newton non-degenerate general polar curve with a fixed Newton polygon. In both cases, we give explicit Zariski open sets the points of which represent branches with Newton non-degenerate polars and describe the topology of their general polars.

3-Fold Extremal Contractions of Types (IC) and (IIB)

Let (X, C) be a germ of a 3-fold X with terminal singularities along an irreducible reduced complete curve C with a contraction f:(X, C) → (Z, o) such that C = f−1 (o)red and −KX is ample. Assume that (X, C) contains a point of type (IC) or (IIB). We complete the classification of such germs in terms of a general member containing C.

Practice and Procedure: Member Schemes

This chapter contains a description of the key documentation that is required for a member scheme of arrangement and examples of a timetable for a general member scheme and a takeover offer that is being effected through a member scheme. It also includes an example of the terms of a scheme for both a topco scheme and a member scheme being used to effect a takeover, together with comments on the key terms of each.

Some projective surfaces of GK-dimension 4

We construct an interesting family of connected graded domains of Gel’fand–Kirillov dimension 4, and show that the general member of this family is noetherian. The algebras we construct are Koszul and have global dimension 4. They fail to be Artin–Schelter Gorenstein, however, showing that a theorem of Zhang and Stephenson for dimension 3 algebras does not extend to dimension 4. The Auslander–Buchsbaum formula also fails to hold for these algebras. The algebras we construct are birational to ℙ2, and their existence disproves a conjecture of the first author and Stafford. The algebras can be obtained as global sections of a certain quasicoherent graded sheaf on ℙ1×ℙ1, and our key technique is to work with this sheaf. In contrast to all previously known examples of birationally commutative graded domains, the graded pieces of the sheaf fail to be ample in the sense of Van den Bergh. Our results thus require significantly new techniques.

Gorenstein Resolutions of 3-Dimensional Terminal Singularities

Let X be a 3-dimensional terminal singularity of index ≥ 2. We shall construct projective birational morphisms ƒ: Y → X such that Y has only Gorenstein terminal singularities and that ƒ factors the minimal resolution of a general member of | −KX |. We also study prime divisors of ƒ, especially the discrepancies of these prime divisors.

On the number of moduli of extendable canonical curves

Let C ⊂ ℙg−1 be a canonical curve of genus g. In this article we study the problem of extendability of C, that is when there is a surface S ⊂ ℙg different from a cone and having C as hyperplane section. Using the work of Epema we give a bound on the number of moduli of extendable canonical curves. This for example implies that a family of large dimension of curves that are cover of another curve has general member nonextendable. Using a theorem of Wahl we prove the surjectivity of the Wahl map for the general k-gonal curve of genus g when k = 5, g ≥ 15 or k = 6, g ≥ 13 or k ≥ 7, g ≥ 12.

ABELIAN VARIETIES OF TYPE III AND THE HODGE CONJECTURE

We show that the algebraicity of Weil's Hodge cycles implies the usual Hodge conjecture for a general member of a PEL-family of abelian varieties of type III. We deduce the general Hodge conjecture for certain 6-dimensional abelian varieties of type III, and the usual Hodge and Tate conjectures for certain 4-dimensional abelian varieties of type III.