Generic hyperplane section of curves and an application to regularity bounds in positive characteristic

This paper investigates the Castelnuovo–Mumford regularity of the generic hyperplane section of projective curves in positive characteristic, and yields an application to a sharp bound on the regularity for nondegenerate projective varieties.     

Stable Maps and Branch Divisors

We construct a natural branch divisor for equidimensional projective morphisms where the domain has lci singularities and the target is nonsingular. The method involves generalizing a divisor construction of Mumford from sheaves to complexes. The construction is valid in flat families. The generalized branch divisor of a stable map to a nonsingular curve X yields a canonical morphism from the space of stable maps to a symmetric product of X. This branch morphism (together with virtual localization) is used to compute the Hurwitz numbers of covers of the projective line for all genera and degrees in terms of Hodge integrals.     

Mumford coverings of the projective line

A Mumford covering of the projective line over a complete non-archimedean valued field K is a Galois covering X? P¹_K X\rightarrow {\bf P}^1_K such that X is a Mumford curve over K. The question which finite groups do occur as Galois group is answered in this paper. This result is extended to the case where P¹_K {\bf P}^1_K is replaced by any Mumford curve over K.     

The abelian monodromy extension property for families of curves

Necessary and sufficient conditions are given (in terms of monodromy) for extending a family of smooth curves over an open subset to a family of stable curves over S. More precisely, we introduce the abelian monodromy extension (AME) property and show that the standard Deligne–Mumford compactification is the unique, maximal AME compactification of the moduli space of curves. We also show that the Baily–Borel compactification is the unique, maximal projective AME compactification of the moduli space of abelian varieties.     

$\Omega$-admissible theory II. Deligne pairings over moduli spaces of punctured Riemann surfaces

In Part I, Deligne-Riemann-Roch isometry is generalized for punctured Riemann surfaces equipped with quasi-hyperbolic metrics. This is achieved by proving the Mean Value Lemmas, which explicitly explain how metrized Deligne pairings for -admissible metrized line bundles depend on . In Part II, we first introduce several line bundles over Knudsen-Deligne-Mumford compactification of the moduli space (or rather the algebraic stack) of stable N-pointed algebraic curves of genus g, which are rather natural and include Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles. Then we use Deligne-Riemann-Roch isomorphism and its metrized version (proved in Part I) to establish some fundamental relations among these line bundles. Finally, we compute first Chern forms of the metrized Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles by using results of Wolpert and Takhtajan-Zograf, and show that the so-called Takhtajan-Zograf metric on the moduli space is algebraic. Received February 14, 2000 / Accepted August 18, 2000 / Published online February 5, 2001     

On the torsion of Jacobians of principal modular curves of level 3<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We demonstrate that the 3-power torsion points of the Jacobians of the principal modular curves X(3ⁿ) are fixed by the kernel of the canonical outer Galois representation of the pro-3 fundamental group of the projective line minus three points. The proof proceeds by demonstrating the curves in question satisfy a two-part criterion given by Anderson and Ihara. Two proofs of the second part of the criterion are provided; the first relies on a theorem of Shimura, while the second uses the moduli interpretation. Received: 30 September 2005     

Differentials of the second kind on mumford curves

We define ap-adic analytic Hodge decomposition for the cohomology of Mumford curves, with values in a local system. When the local system is trivial, we give a new proof of Gerritzen’s theorem, that this decomposition forms a variation of Hodge structure, in a family of Mumford curves.     

Generalized Poincaré Classes and Cubic Equivalences

Let Y be a smooth projective algebraic surface over ?, and T(Y) the kernel of the Albanese map CH₀(Y)_deg0 → Alb(Y). It was first proven by D. Mumford that if the genus Pg(Y) > 0, then T(Y) is 'infinite dimensional'. One would like to have a better idea about the structure of T(Y). For example, if Y is dominated by a product of curves E₁ × E₂, such as an abelian or a Kummer surface, then one can easily construct an abelian variety B and a surjective 'regular' homomorphism B^?z2 → T(Y). A similar story holds for the case where Y is the Fano surface of lines on a smooth cubic hypersurface in P⁴. This implies a sort of boundedness result for T(Y). It is natural to ask if this is the case for any smooth projective algebraic surface Y ? Partial results have been attained in this direction by the author [Illinois. J. Math. 35 (2), 1991]. In this paper, we show that the answer to this question is in general no. Furthermore, we generalize this question to the case of the Chow group of k—cycles on any projective algebraic manifold X, and arrive at, from a conjectural standpoint, necessary and sufficient cohomological conditions on X for which the question can be answered affirmatively.     

Riemann existence theorems of Mumford type

Riemann Existence Theorems for Galois covers of Mumford curves by Mumford curves are stated and proven. As an application, all finite groups are realised as full automorphism groups of Mumford curves in characteristic zero.     

Some Koszul rings from geometry

We give examples of Koszul rings that arise naturally in algebraic geometry. In the first part, we prove a general result on Koszul property associated to an ample line bundle on a projective variety. Specifically, we show how Koszul property of multiples of a base point free ample line bundle depends on its Castelnuovo–Mumford regularity. In the second part, we give examples of Koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type.     

Vers un algorithme pour la réduction stable des revêtements <Emphasis Type="Italic">p</Emphasis>-cycliques de la droite projective sur un corps <Emphasis Type="Italic">p</Emphasis>-adique

 In his Ph. D. thesis, C. Lehr offers an algorithm which gives the stable model for p-cyclic covers of the projective line over a p-adic field under the conditions that the branch locus whose cardinal is m+1 has the so called equidistant geometry and m<p. In this note we give an algorithm also in the equidistant geometry case but without condition on m. In particular we are able to study the reduction at 2 of hyperelliptic curves with equidistant branch locus.

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Vers un algorithme pour la réduction stable des revêtements p-cycliques de la droite projective sur un corps p-adique

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Received: 11 February 2002 / Revised version: 8 May 2002 / Published online: 2 December 2002

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Mathematics Subject Classification (2000): 11G20, 14H30, 14Q05     

Construction techniques for Galois coverings of the affine line

For constructing un ramified coverings of the affine line in characteristicp, a general theorem about good reductions modulop of coverings of characteristic zero curves is proved. This is applied to modular curves to realize SL(2, ℤ/nℤ)/±1, with GCD(n, 6) = 1, as Galois groups of unramified coverings of the affine line in characteristicp, for p = 2 or 3. It is applied to the Klein curve to realize PSL(2, 7) for p = 2 or 3, and to the Macbeath curve to realize PSL(2, 8) for p = 3. By looking at curves with big automorphism groups, the projective special unitary groups PSU(3, p^v) and the projective special linear groups PSL(2, p^v) are realized for allp, and the Suzuki groups Sz(2^2v+1) are realized for p = 2. Jacobian varieties are used to realize certain extensions of realizable groups with abelian kernels.     

A fake projective plane with an order 7 automorphism

A fake projective plane is a compact complex surface (a compact complex manifold of dimension 2) with the same Betti numbers as the complex projective plane, but not isomorphic to the complex projective plane. As was shown by Mumford, there exists at least one such surface.In this paper we prove the existence of a fake projective plane which is birational to a cyclic cover of degree 7 of a Dolgachev surface.     

Rationally connected varieties—On a conjecture of Mumford

We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.     

Inversion of degree <Emphasis Type="Italic">n</Emphasis> + 2

By the method of synthetic geometry, we define a seemingly new transformation of a three-dimensional projective space where the corresponding points lie on the rays of the first order, nth class congruence C _n ¹ and are conjugate with respect to a proper quadric Ψ. We prove that this transformation maps a straight line onto an n + 2 order space curve and a plane onto an n + 2 order surface which contains an n-ple (i.e. n-multiple) straight line. It is shown that in the Euclidean space the pedal surfaces of the congruences C _n ¹ can be obtained by this transformation. The analytical approach enables new visualizations of the resulting curves and surfaces with the program Mathematica. They are shown in four examples.      

On the Semistability of Instanton Sheaves Over Certain Projective Varieties

We show that instanton bundles of rank r ≤ 2n ? 1, defined as the cohomology of certain linear monads, on an n-dimensional projective variety with cyclic Picard group are semistable in the sense of Mumford–Takemoto. Furthermore, we show that rank r ≤ n linear bundles with nonzero first Chern class over such varieties are stable. We also show that these bounds are sharp.     

The cone of pseudo-effective divisors of log varieties after Batyrev

In these notes, we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective \mathbb Q{\mathbb Q}-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev’s structure theorem for the cone of nef curves of projective terminal threefolds.     

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ? (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over that lifts to ℤ/p⁷ but not ℤ/p⁸. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mn?v’s universality theorem. Mathematics Subject Classification (2000) 14B12, 14C05, 14J10, 14H50, 14B07, 14N20, 14D22, 14B05     

Galois lines for the Artin–Schreier–Mumford curve

The arrangement of all Galois lines for the Artin–Schreier–Mumford curve in the projective 3-space is described. Surprisingly, there exist infinitely many Galois lines intersecting this curve.     

On rational embeddings of curves in the second Garcia–Stichtenoth tower

In 1995, Garcia and Stichtenoth explicitly constructed a tower of projective curves over a finite field with q² elements which reaches the Drinfeld–Vlăduţ bound. These curves are given recursively by covers of Artin–Schreier type where the curve on the nth level of the tower has a natural model in . In this paper, for q an even prime power, we use point projections in order to embed these curves into projective space of the lowest possible dimension.