Abstract
Given a smooth cubic hypersurface X over a finite field of characteristic greater than 3 and two generic points on X, we use a function field analogue of the Hardy–Littlewood circle method to obtain an asymptotic formula for the number of degree d k-rational curves on X passing through those two points. We use this to deduce the dimension and irreducibility of the moduli space parametrising such curves, for large enough d.
Highlights
Where fi ∈ k[u, v] are homogeneous polynomials of degree d 1, with no non-constant common factor in k[u, v], such that
In [13, Example 7.6], Kollar proves that there exists a constant cn depending only on n such that for any q > cn and any point x ∈ X(k), there exists a k-rational curve of degree at most 216 on X passing through x
In our investigation we focus on the case m = 2 of 2-pointed rational curves on X
Summary
We establish notation and record some basic definitions and facts. Throughout this paper S ≪ T denotes an estimate of the form S CT , where C is some constant that does not depend on q. Finite primesin O are monic irreducible polynomials and we let s = t−1 be the prime at infinity. These have associated absolute values which extend to give absolute values | · |̟ and | · | = | · |∞ on K. The three results are standard, but are proved here since we require versions in which the implied constant is independent of q. Let τ (f ) be the number of monic divisors of a polynomial f ∈ Fq[t]. Let ω(f ) denote the number of prime divisors of a polynomial f ∈ Fq[t]. Let τk(f ) denote the number of factorisations of a polynomial f ∈ Fq[t] into k factors.
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