Curves In Positive Characteristic Research Articles

Galois covers of singular curves in positive characteristics

Galois covers of singular curves in positive characteristics

Finiteness of leaps in the sense of Hasse-Schmidt of unibranch curves in positive characteristic

We prove that the set of leaps of the chain of m-integrable derivations of a curve X over a perfect field with geometrically unibranch singularities is finite. This result is a consequence of an affirmative answer to Seidenberg's question of extending, in positive characteristic, Hasse–Schmidt derivations of finite length of the local rings of X to their integral closures.

Non-abelian cohomology of universal curves in positive characteristic

In this paper, we will compute the non-abelian cohomology of the universal complete curve in positive characteristic. This extends Hain’s result on the non-abelian cohomology of generic curves in characteristic zero to positive characteristics. Furthermore, we will prove that the exact sequence of etale fundamental groups of the universal n n -punctured curve in positive characteristic does not split.

Factoring differential operators over algebraic curves in positive characteristic

We present an algorithm for factoring linear differential operators with coefficients in a finite separable extension of F p ( x ). Our methods rely on specific tools arising in positive characteristic: p -curvature, structure of simple central algebras and p -Riccati equations.

Asymptotic socle behaviors for cones over curves in positive characteristic

This paper studies the distribution of socle degrees of $R/I^{[p^e]}$ when $e$ is large, for a homogeneous ideal $I$ in a two-dimensional standard-graded normal domain $R$ in positive characteristic $p$. We prove that the distribution is very much related to the asymptotic slopes of the syzygy bundle Syz$(I)$, which have been known to determine the Hilbert-Kunz multiplicity of $I$.

Deformations of rational curves in positive characteristic

Abstract We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic p is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than 1 2 ⁢ ( p - 1 ) {\frac{1}{2}(p-1)} (resp. p), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher-dimensional varieties. Moreover, we show by example that our result is in some sense optimal. On our way, we obtain a sufficient criterion in terms of Jacobian numbers for the normalization of a curve over an imperfect field to be smooth.

Rational points of universal curves in positive characteristics

Let K K be the function field of the moduli stack M g , n / F p {\mathcal M}_{g,n/{\mathbb F}_p} of curves over Spec F p \text {Spec}{\mathbb F}_p , and let C / K C/K be the restriction of the universal curve to Spec K \text {Spec} K . We show that if g ≥ 3 g\geq 3 , then the only K K -rational points of C C are its n n tautological points. Furthermore, we show that if g ≥ 4 g\geq 4 and n = 0 n=0 , then Grothendieck’s section conjecture holds for C C over K K . This extends Hain’s work in characteristic 0 0 to positive characteristics.

On the averages of generalized Hasse–Witt invariants of pointed stable curves in positive characteristic

In the present paper, we study fundamental groups of curves in positive characteristic. Let $$X^{\bullet }$$ be a pointed stable curve of type $$(g_{X}, n_{X})$$ over an algebraically closed field of characteristic $$p>0$$, $$\Gamma _{X^{\bullet }}$$ the dual semi-graph of $$X^{\bullet }$$, and $$\Pi _{X^{\bullet }}$$ the admissible fundamental group of $$X^{\bullet }$$. In the present paper, we study a kind of group-theoretical invariant $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ associated to the isomorphism class of $$\Pi _{X^{\bullet }}$$ called the limit of p-averages of $$\Pi _{X^{\bullet }}$$, which plays a central role in the theory of anabelian geometry of curves over algebraically closed fields of positive characteristic. Without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, we give a lower bound and a upper bound of $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$. In particular, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ under a certain assumption concerning $$\Gamma _{X^{\bullet }}$$ which generalizes a formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ obtained by Tamagawa. Moreover, if $$X^{\bullet }$$ is a component-generic pointed stable curve, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, which can be regarded as an averaged analogue of the results of Nakajima, Zhang, and Ozman–Pries concerning p-rank of abelian etale coverings of projective generic curves for admissible coverings of component-generic pointed stable curves.

Cyclic coverings of the projective line by Mumford curves in positive characteristic

We study the rigid analytic geometry of cyclic coverings of the projective line. We determine the defining equation of cyclic coverings of degree $p$ of the projective line by Mumford curves over complete discrete valuation fields of positive characteristic $p$. Previously, Bradley studied that of any degree over non-archimedean local fields of characteristic zero.

Simple singularities of parametrized plane curves in positive characteristic

ABSTRACTLet K be an algebraically closed field of characteristic p>0. The aim of the article is to give a classification of simple parametrized plane curve singularities over K. The idea is to give explicitly a class of families of singularities which are not simple such that almost all singularities deform to one of those and show that remaining singularities are simple.

Gonality of abstract modular curves in positive characteristic

Let$C$be a smooth, separated and geometrically connected curve over a finitely generated field$k$of characteristic$p\geqslant 0$,$\unicode[STIX]{x1D702}$the generic point of$C$and$\unicode[STIX]{x1D70B}_{1}(C)$its étale fundamental group. Let$f:X\rightarrow C$be a smooth proper morphism, and$i\geqslant 0$,$j$integers. To the family of continuous$\mathbb{F}_{\ell }$-linear representations$\unicode[STIX]{x1D70B}_{1}(C)\rightarrow \text{GL}(R^{i}f_{\ast }\mathbb{F}_{\ell }(j)_{\overline{\unicode[STIX]{x1D702}}})$(where$\ell$runs over primes$\neq p$), one can attach families of abstract modular curves$C_{0}(\ell )$and$C_{1}(\ell )$, which, in this setting, are the analogues of the usual modular curves$Y_{0}(\ell )$and$Y_{1}(\ell )$. If$i\not =2j$, it is conjectured that the geometric and arithmetic gonalities of these abstract modular curves go to infinity with$\ell$(for the geometric gonality, under a certain necessary condition). We prove the conjecture for the arithmetic gonality of the abstract modular curves$C_{1}(\ell )$. We also obtain partial results for the growth of the geometric gonality of$C_{0}(\ell )$and$C_{1}(\ell )$. The common strategy underlying these results consists in reducing by specialization theory to the case where the base field$k$is finite in order to apply techniques of counting rational points.

Variation of fundamental groups of curves in positive characteristic

In this paper we prove that given a non-isotrivial family of hyperbolic curves in positive characteristic, the isomorphism type of the geometric fundamental group is not constant on the fibres of the family.

Erratum to: Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic

Let k be an algebraically closed field of characteristic p > 0 of countable cardinality. Denote by C a smooth, connected, affine curve defined over k with function field K and by π1(C) its algebraic fundamental group. In the original paper we addressed the question of the structure of the normal closed subgroups of π1(C). More precisely, in Theorem 1.1, we gave a sufficient condition for such a normal closed subgroup N to be isomorphic to a subgroup of a free profinite group of countable rank. There we claimed to prove this under the hypothesis that the group M = π1(C)/N had an infinitely generated Sylow p-subgroup. However, as remarked in the paragraph after Corollary 1.2 of [1], this hypothesis is not enough. We would like to make this remark more explicit by stating (as noticed by those authors) the additional hypothesis needed to make Theorem 1.1 work. For each integer g ≥ 0 denote by Pg(C) the intersection of algebraic fundamental groups

The fundamental group of affine curves in positive characteristic

It is shown that the commutator subgroup of the fundamental group of a smooth irreducible affine curve over an uncountable algebraically closed field k of positive characteristic is a profinite free group of rank equal to the cardinality of k.

Deformations of elliptic fiber bundles in positive characteristic

AbstractWe study the deformation theory of elliptic fiber bundles over curves in positive characteristics. As applications, we give examples of nonliftable elliptic surfaces in characteristics 2 and 3, which answer a question of Katsura and Ueno. Also, we construct a class of elliptic fibrations, whose liftability is equivalent to a conjecture of Oort concerning the liftability of automorphisms of curves. Finally, we classify deformations of bielliptic surfaces.

Deformations of elliptic fiber bundles in positive characteristic

AbstractWe study the deformation theory of elliptic fiber bundles over curves in positive characteristics. As applications, we give examples of nonliftable elliptic surfaces in characteristics 2 and 3, which answer a question of Katsura and Ueno. Also, we construct a class of elliptic fibrations, whose liftability is equivalent to a conjecture of Oort concerning the liftability of automorphisms of curves. Finally, we classify deformations of bielliptic surfaces.

On the ring of unipotent vector bundles on elliptic curves in positive characteristics

Using Fourier�Mukai transforms, we prove some results about the ring of unipotent vector bundles on elliptic curves in positive characteristics. This ring was determined by Atiyah in characteristic zero, who showed that it is a polynomial ring in one variable. It turns out that the situation in characteristic p > 0 is completely different and rather bizarre: the ring is nonnoetherian and contains a subring whose spectrum contains infinitely many copies of Spec(Z), which are glued with successively higher and higher infinitesimal identification at the point corresponding to the prime p.

On the generic plane section of smooth space curves in positive characteristic

Let C ⊂ P3 be a smooth and connected projective curve of genus g ≥ 2. Here in characteristic ≠ 2 we prove that its generic plane section is in uniform position and that its numerical character is connected.

Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic

Let π1(C) be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic p > 0 of countable cardinality. Let N be a normal (respectively, characteristic) subgroup of π 1(C). Under the hypothesis that the quotient π 1(C)/N admits an infinitely generated Sylow p-subgroup, we prove that N is indeed isomorphic to a normal (respectively, characteristic) subgroup of a free profinite group of countable cardinality. As a consequence, every proper open subgroup of N is a free profinite group of countable cardinality.

On the ample vector bundles over curves in positive characteristic

Let E be an ample vector bundle over a smooth projective curve defined over an algebraically closed field of positive characteristic. We construct a family of curves in the total space of E, parametrized by an affine space, that surjects onto the total space of E and give a deformation of (nonreduced) zero section of E. To cite this article: I. Biswas, A.J. Parameswaran, C. R. Acad. Sci. Paris, Ser. I 339 (2004).