On the proportion of transverse-free plane curves

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Abstract

We study the asymptotic proportion of smooth plane curves over a finite field Fq which are tangent to every line defined over Fq. This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver's theorem on perfect matchings in regular bipartite graphs. Our main theorem implies that a random smooth plane curve over Fq admits a transverse Fq-line with very high probability.

Introduction

Let CP2 be a plane curve defined over an arbitrary field k. If k is an infinite field, then we can apply Bertini's theorem to find a line L defined over k such that L meets C transversely. By Bézout's theorem, any line (which is not a component of the curve C) meets C in exactly d=deg(C) points with coordinates in k counted with multiplicity; by definition, a transverse line is one which meets C in d distinct points. Equivalently, L is not transverse to C if and only if L is tangent to C or L passes through a singular point of C.

When k=Fq is a finite field, it is not necessarily true that we can find a line L defined over Fq such that L meets C transversely. It is possible that each of the q2+q+1 lines in P2 defined over Fq is tangent to C. Indeed, there exist smooth plane curves of degree q+2 with this property [1, Example 2.A]. There are also smooth examples in each sufficiently large degree [1, Example 2.B].

Given a plane curve CP2 defined over a finite field Fq, we say that C is transverse-free if C admits no transverse line defined over Fq.

In the present paper, we will restrict our attention to smooth plane curves. In order to count the proportion of smooth transverse-free plane curves, we use the notion of a natural density denoted by μ. Given a subset of AR:=Fq[x,y,z], the natural density μ(A) is the limit as d of the fraction of elements of degree d in A among all degree d homogeneous polynomials in R. Note that counting homogeneous polynomials and curves of the same degree differ by a factor arising from scaling, so that the limiting proportion for both counts coincide. Our main result is the following.

Theorem 1.1

Let FFq[x,y,z] be the set of polynomials defining smooth transverse-free plane curves. Thene(q2+q+1)(1q+1q21q3)q2+q+1μ(F)152(1q+1q21q3)q2+q+1

where e=2.71828... is Euler's number.

Acknowledgments. We are grateful to Charles Favre who first raised the question of computing the natural density of transverse-free plane curves over finite fields. C. Favre asked for the exact value of μ(F), and Theorem 1.1 is our partial answer. We also thank Dori Bejleri and Zinovy Reichstein for helpful conversations on this topic. Both authors were supported by postdoctoral research fellowships from the University of British Columbia. The second author was partially supported by a postdoctoral fellowship from The Pacific Institute for the Mathematical Sciences.

Section snippets

Overview of the paper

In this section, we discuss notation used in the paper, interpret our main theorem, and explain how the paper is organized.

We will work with the finite fields Fq where q is a fixed prime power. We set R:=Fq[x,y,z] and denote by Rd the vector space spanned by degree d homogeneous polynomials in R. By convention, 0 is considered a homogeneous polynomial in each degree. As a shorthand, we also defineRhomog=d1Rd to be the set of non-constant homogeneous polynomials. By definition, elements of R

Proof of the lower bound

The objective of this section is to prove the following.

Theorem 3.1

Let FR=Fq[x,y,z] denote the subset defining smooth transverse-free plane curves. Thenμ_(F)e(q2+q+1)(1q+1q21q3)q2+q+1.

Our task is to produce a positive proportion of smooth transverse-free curves of degree d. Recall that a smooth plane curve C is transverse-free if each of the q2+q+1 lines defined over Fq is tangent to C at some geometric point (i.e. a point defined over Fq). By requiring each of the tangency points to be an Fq-point

Computing natural density with examples

In this section, we investigate the natural densities of various subsets of Rhomog. These results will be used in Section 5.

The following three subsets and their properties will be the focus of this section. The first few lemmas will compute the densities of these subsets. Given a point QP2(Fq), we defineSQ={fRhomog|C={f=0} is singular at Q}. Given a line L and a point PP2(Fq) with PL, we defineTL,P={fRhomog|C={f=0} is tangent to L at P}. Note that the condition that C is tangent to L at P

Proof of the upper bound

In this section, our goal is to prove the following result.

Theorem 5.1

Let FR=Fq[x,y,z] denote the subset defining smooth transverse-free plane curves. Thenμ(F)(1q2)(q2+q+1)(1q+1q21q3)q2+q+1.

Let us explain how the upper bound in Theorem 5.1 is stronger than the upper bound given in Theorem 1.1. It suffices to show that(1q2)(q2+q+1)152 for all q2. The function ξ(q):=(1q2)(q2+q+1) is decreasing for q2, and in fact, limqξ(q)=e. In particular, we have ξ(q)ξ(2). Consequently,ξ(q)=(1q2)(q

Existence of the natural density

In this section, we prove that μ(F) exists as a limit. As a consequence, μ_(F)=μ(F)=μ(F).

Proposition 6.1

The subsetF={fRhomog|C={f=0}is a smooth plane curvesuch that all Fq-lines are tangent to C} has a well-defined natural density μ(F).

Proof

As explained in Section 5.1, we can writeF=r=1Fr. We will show that μ(Fr) exists for each r1. Let us first explain how this will prove the proposition. The sequence (μ(Fr))r=1 is clearly increasing and also bounded by Theorem 5.2, and hence converges. Using inequality

Declaration of Competing Interest

None.

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