On the proportion of transverse-free plane curves
Introduction
Let 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 points with coordinates in 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 is a finite field, it is not necessarily true that we can find a line L defined over such that L meets C transversely. It is possible that each of the lines in defined over is tangent to C. Indeed, there exist smooth plane curves of degree 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 defined over a finite field , we say that C is transverse-free if C admits no transverse line defined over .
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 , the natural density is the limit as of the fraction of elements of degree d in 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 be the set of polynomials defining smooth transverse-free plane curves. Then
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 , 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 where q is a fixed prime power. We set and denote by 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 define to be the set of non-constant homogeneous polynomials. By definition, elements of
Proof of the lower bound
The objective of this section is to prove the following.
Theorem 3.1 Let denote the subset defining smooth transverse-free plane curves. Then
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 lines defined over is tangent to C at some geometric point (i.e. a point defined over ). By requiring each of the tangency points to be an -point
Computing natural density with examples
In this section, we investigate the natural densities of various subsets of . 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 , we define Given a line L and a point with , we define 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 denote the subset defining smooth transverse-free plane curves. Then
Existence of the natural density
In this section, we prove that exists as a limit. As a consequence, .
Proposition 6.1 The subset has a well-defined natural density .
Proof As explained in Section 5.1, we can write We will show that exists for each . Let us first explain how this will prove the proposition. The sequence is clearly increasing and also bounded by Theorem 5.2, and hence converges. Using inequality
Declaration of Competing Interest
None.
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