Monodromy of Hyperplane Sections of Curves and Decomposition Statistics over Finite Fields

Abstract

For a projective curve⊂ defined over we study the statistics of the -structure of a section of by a random hyperplane defined over in the → ∞ limit. We obtain a very general equidistribution result for this problem. We deduce many old and new results about decomposition statistics over finite fields in this limit. Our main tool will be the calculation of the monodromy of transversal hyperplane sections of a projective curve.

Sectional monodromy groups of projective curves

“…Rathmann also gave an example of a smooth rational curve in with . Rathmann's theorem is important for understanding statistics over function fields, for example the large characteristic version of the Bateman–Horn conjecture (see [14, Section 6] for examples of such statistical problems). In Section 3, we prove an extension of Rathmann's theorem that applies to nonsmooth curves in .…”

Factorization type probabilities of polynomials with prescribed coefficients over a finite field

“…Let U be a dense open subset of A 1 b such that ϕ s | ϕ −1 s (U ) : ϕ −1 s (U) → U is finite andétale. The statement now follows from the Chebotarev density theorem for function fields (see Theorem 3 in [3]) applied to ϕ s | ϕ −1 s (U ) .…”

“…The proof is based on the technique employed by Entin in a variety of problems solved in [3], with an extra ingredient (Lemma 5 below) developed by the author in an earlier work, concerning the irreducibility of the perturbations of a certain curve. Namely, for s ∈ F q , we set up a genericallyétale map ϕ s : Ω s → A 1 b of degree d between geometrically irreducible varieties over F q such that for any b ∈ A 1 (F q ) with d preimages over F q , the conjugacy class in S d that the action of the Frobenius Fr q on ϕ −1 s (b) gives rise to has cycle structure corresponding to the factorization type of the polynomial f (T ) + sT + b ∈ F q [T ].…”

Motivic limits for Fano varieties of $k$-planes

“…In Theorem 1.7, the connection of F q -point counts with Remark ?? is that the monodromy action induced by the covering map J −→ G(n − m, n) could be involved in studying how the point count above is distributed among conjugacy classes of the action of Frobenius on general (n − m)-plane sections of Y (see [11]).…”