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Register a new userNormal subgroups of fundamental groups of affine curves in positive characteristic II
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Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C/k$ be a smooth connected affine curve. Denote by $pi_1(C)$ its algebraic fundamental group. The goal of this paper is to characterize a certain subset of closed normal subgroups $N$ of $pi_1(C)$. In Normal subgroups of fundamental groups of affine curves in positive characteristic we proved the same result under the additional hypothesis that $k$ had countable cardinality.
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Let $pi_1(C)$ 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 (resp. characteristic) subgroup of $pi_1(C)$. Under the hypothesis that the quotient $pi_1(C)/N$ admits an infinitely generated Sylow $p$-subgroup, we prove that $N$ is indeed isomorphic to a normal (resp. 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.
We study the normal map for plane projective curves, i.e., the map associating to every regular point of the curve the normal line at the point in the dual space. We first observe that the normal map is always birational and then we use this fact to show that for smooth curves of degree higher than four the normal map uniquely determines the curve. Our proof works in characteristic zero and in positive characteristic higher than the degree of the curve. We notice also that in high characteristic strange curves provide examples of different plane curves with same curve of normal lines. We will reinterpret our results also in the modern terminology of bottlenecks of algebraic curves.
The first main purpose of this paper is to contribute to the existing knowledge about the complex projective surfaces $S$ of general type with $p_g(S) = 0$ and their moduli spaces, constructing 19 new families of such surfaces with hitherto unknown fundamental groups. We also provide a table containing all the known such surfaces with K^2 <=7. Our second main purpose is to describe in greater generality the fundamental groups of smooth projective varieties which occur as the minimal resolutions of the quotient of a product of curves by the action of a finite group. We classify, in the two dimensional case, all the surfaces with q=p_g = 0 obtained as the minimal resolution of such a quotient, having rational double points as singularities. We show that all these surfaces give evidence to the Bloch conjecture.
We prove Clifford theoretic results on the representations of finite groups which only hold in characteristic $2$. Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let $varphi$ be an irreducible $2$-Brauer character of $N$ which is self-dual. We prove that there is a unique self-dual irreducible Brauer character $theta$ of $G$ such that $varphi$ occurs with odd multiplicity in the restriction of $theta$ to $N$. Moreover this multiplicity is $1$. Conversely if $theta$ is an irreducible $2$-Brauer character of $G$ which is self-dual but not of quadratic type, the restriction of $theta$ to $N$ is a sum of distinct self-dual irreducible Brauer character of $N$, none of which have quadratic type. Let $b$ be a real $2$-block of $N$. We show that there is a unique real $2$-block of $G$ covering $b$ which is weakly regular.
We describe examples motivated by the work of Serre and Abhyankar.
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