arXiv subjects

Aaron Michael Silberstein

Publications and source records attributed to Aaron Michael Silberstein.

3 recordsLinked to original sources

arXiv2016

Families of Disjoint Divisors on Varieties

Following the work of Totaro and Pereira, we study sufficient conditions under which collections of pairwise-disjoint divisors on a variety over an algebraically closed field are contained in the fibers of a morphism to a curve. We prove that $ρ_w(X) + 1$ pairwise-disjoint, connected divisors suffices for proper, normal varieties $X$, where $ρ_w(X)$ is a modification of the Néron-Severi rank of $X$ (they agree when $X$ is projective and smooth). We then prove a strong counterexample in the affine case: if $X$ is quasi-affine and of dimension $\geq 2$ over a countable, algebraically-closed field $k$, then there exists a (countable) collection of pairwise-disjoint divisors which cover the $k$-points of X, so that for any non-constant morphism from $X$ to a curve, at most finitely many are contained in the fibers thereof. We show, however, that an uncountable collection of pairwise-disjoint, connected divisors in any normal variety over an algebraically-closed field must be contained in the fibers of a morphism to a curve.

math.AG↗

arXiv2013

Anabelian Intersection Theory I: The Conjecture of Bogomolov-Pop and Applications

We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let $F_{1}$ and $F_{2}$ be fields finitely-generated and of transcendence degree $\geq 2$ over $k_{1}$ and $k_{2}$, respectively, where $k_{1}$ is either $\bar{\mathbb{Q}}$ or $\bar{\mathbb{F}}_{p}$, and $k_{2}$ is algebraically closed. We denote by $G_{F_1}$ and $G_{F_2}$ their respective absolute Galois groups. Then the canonical map $φ_{F_{1}, F_{2}}: \Isom^i(F_1, F_2)\rightarrow \Isom^{\Out}_{\cont}(G_{F_2}, G_{F_1})$ from the isomorphisms, up to Frobenius twists, of the inseparable closures of $F_1$ and $F_2$ to continuous outer isomorphisms of their Galois groups is a bijection. Thus, function fields of varieties of dimension $\geq 2$ over algebraic closures of prime fields are anabelian. We apply this to give a necessary and sufficient condition for an element of the Grothendieck-Teichmüller group to be an element of the absolute Galois group of $\bar{\mathbb{Q}}$.

math.AG↗

arXiv2009

Representations of Galois Groups on the Homology of Surfaces

Let $p:Σ'\toΣ$ be a finite Galois cover, possibly branched, with Galois group $G$. We are interested in the structure of the cohomology of $Σ'$ as a module over $G$. We treat the cases of branched and unbranched covers separately. In the case of branched covers, we give a complete classification of possible module structures of $H_1(Σ',\bC)$. In the unbranched case, we algebro-geometrically realize the representation of $G$ on holomorphic 1-forms on $Σ'$ as functions on unions of $G$-torsors. We also explicitly realize these representations for certain branched covers of hyperelliptic curves. We give some applications to the study of pseudo-Anosov homeomorphisms of surfaces and representation theory of the mapping class group.

math.GT↗