Rational curves on cubic hypersurfaces over finite fields

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 $d$ 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 $d$.


Introduction
Let k = F q be a finite field and let F ∈ k[x 1 , . . . , x n ] denote a non-singular homogeneous polynomial of degree 3. Moreover, let X ⊂ P n−1 k be the smooth cubic hypersurface defined over k by F = 0. Let C be a smooth projective curve over k. Then k(C) has transcendence degree 1 over a C 1 -field and, by the Lang-Tsen theorem [6,Theorem 3.6], the set X(k(C)) of k(C)-rational points on X is non-empty for n 10. This still holds for X singular. In this paper we are interested in the case C = P 1 , writing K = k(t) denote the function field of C over k. A degree d rational curve on X is a non-constant morphism f : P 1 k → X given by f = (f 1 (u, v), . . . , f n (u, v)) , (1.1) where f i ∈k [u, v] are homogeneous polynomials of degree d 1, with no non-constant common factor ink [u, v], such that F (f 1 (u, v), . . . , f n (u, v)) ≡ 0.
Such a curve is said to be m-pointed if it is equipped with a choice of m distinct points P 1 , . . . , P m ∈ X(k) called the marks through which the curve passes. Up to isomorphism, these curves are parametrised by the moduli space M 0,m (P 1 k , X, d). The compactification of this space, M 0,m (P 1 k , X, d), is the Kontsevich moduli space of stable maps.
Suppose from now on that #k = q and char(k) > 3. In [13,Example 7.6], Kollár proves that there exists a constant c n depending only on n such that for any q > c n 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.
Associate to F the Hessian matrix for the corresponding points in X(k). As is well-known (see [9,Lemma 1], for example), the Hessian H(x) does not vanish identically on X, since char(k) > 3. The main goal of this paper is to obtain an asymptotic formula for the number of rational curves of degree d on X passing through a and b. Denote the space of such curves by Mor d,a,b (P 1 k , X). We can write the f i in (1.1) explicitly as where α (i) j ∈ k for 0 j d and 1 i n. Then, we capture the condition that the rational curve f passes through the points a and b by selecting  There exists a correspondence between the rational curves on X of bounded degree and the K-points on X of bounded height. Define N a,b (d) to be the number of polynomials f 1 , . . . , f n ∈ F q [t] of degree at most d whose constant coefficients are given by a and whose leading coefficients are given by b, such that F (f 1 , . . . , , f n ) = 0. Thus, N a,b (d) counts the F q -points (f 1 , . . . , f n ) on the affine cone of Mor d,a,b (P 1 k , X), where the condition that f 1 , . . . , f n have no common factor is dropped. Using a version of the Hardy-Littlewood circle method for the function field K developed by Lee [18,19], and further by Browning-Vishe [1], we shall obtain the following result. Theorem 1.1. Fix k = F q with char(k) > 3. Fix a smooth cubic hypersurface X ⊂ P n−1 k , where n 10. Let a, b ∈ X(k), not both on the Hessian. Then, we have where the implied constant in the estimate depends only on d and X.
The condition that one of the two fixed points is not on the Hessian comes from our analysis of certain oscillatory integrals (see Lemma 3.5).
Although it would be possible to generalise Theorem 1.1 to handle rational curves passing through any generic finite set of points in X(k), the main motivation for considering rational curves through two fixed points comes from the notion of rational connectedness. In [21], Manin defined R-equivalence on the set of rational points of a variety in order to study the parametrisation of rational points on cubic surfaces. We say that two points a, b ∈ X(k) are directly R-equivalent if there is a morphism f : P 1 → X (defined over k) with f (0, 1) = a and f (1, 0) = b; the generated equivalence relation is called R-equivalence. In [25], Swinnerton-Dyer proved that R-equivalence is trivial on smooth cubic surfaces over finite fields; that is, all k-points are Requivalent. Next, the result was generalised for smooth cubic hypersurfaces X ⊂ P n−1 k , if n 6 by Madore in [20], and if n 4 and q 11 by Kollár in [13]. Moreover, Madore's result holds for X defined over any C 1 field. The study of R-equivalence is closely related to understanding the geometry of the moduli space of rational curves. In particular, it is interesting to study Requivalence in the case of varieties with many rational curves. Such varieties are called rationally connected and were first studied by Kollár, Miyaoka and Mori in [14], and independently by Campana in [3]. Roughly speaking, Y is rationally connected if for two general points of Y there is a rational curve on Y passing through them. Thus, rationally connected varieties are varieties for which R-equivalence becomes trivial when one extends the ground field to an arbitrary algebraically closed field. Note that in the case of fields of positive characteristic one should consider separably rationally connected varieties. For precise definitions and a thorough introduction to the theory see Kollár [11], [12], and Kollár-Szabó [15].
Fix a smooth cubic hypersurface X ⊂ P n−1 k , where n 10. Then there exists a constant c X > 0 such that for any points a, b ∈ X(k), not both on the Hessian, and any d 19(n−1) n−9 , if q c X , then there exists a F q -rational curve C ⊂ X of degree d that passes through a and b.
This can also be seen as a corollary of Pirutka [22,Proposition 4.3] which states that any two points a, b ∈ X(k) can be joined by two lines on X defined over k.
Keeping track of the dependance on q allows us to deduce further results regarding the geometry of the moduli space Mor d,a,b (P 1 k , X), in the spirit of those obtained by Browning-Vishe [2]. We can regard f in (1.1) under the conditions given by (1.2) as a point in P    [17], we obtain that the space Mor d,a,b (P 1 k , X) is irreducible and of expected dimension µ. Following the same "spreading out" argument (see [5, §10.4.11] and [24]) as in [2, §2], the problem over C can be related to the problem over F q , and this leads to the following corollary. Corollary 1.4. Fix a smooth cubic hypersurface X ⊂ P n−1 defined over C, where n 10. Pick any points two points in X(C), not both on the Hessian. Then for each d 19(n−1) n−9 , the space M 0,2 (P 1 C , X, d) is irreducible and of expected dimensionμ = µ − 3.
In the case of stable maps, Harris-Roth-Starr [7] prove that for a general hypersurface X ⊂ P n−1 C of degree at most n − 2, the Kontsevich moduli space M 0,m (P 1 C , X, d) is a generically smooth, irreducible local complete intersection stack of the expected dimension.
Acknowledgements. I would like to thank my supervisor Tim Browning for suggesting the problem and for all the useful discussions. The author is supported by an EPSRC doctoral training grant.

Preliminaries
In this section 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. Similarly, the implied constants in the notation S = O(T ) are independent of q. Let k = F q , K = k(t), and O = k[t]. Finite primes ̟ in 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. We let K ̟ and K ∞ be the completions. We have Locally compact topological spaces have Haar measures, hence there is a (Haar) measure on K ∞ , and so on T. This is normalised such that T dα = 1 and is extended to K ∞ in such a way that for any positive integer N. Moreover, this can be extended to T n and K n ∞ for any n ∈ Z >0 . Denote by ψ : K ∞ → C * the non-trivial additive character on K ∞ , given by where q is a power of p.Throughout this paper, for any real number R, let R = q R . The following orthogonality property in [16,Lemma 7] holds.
Lemma 2.1. For any N ∈ Z 0 and any γ ∈ K ∞ , we have The following lemma corresponds to [1, Lemma 2.2] and a proof can also be found in [16, Lemma 1(f)].
The next three results are standard, but are proved here since we require versions in which the implied constant is independent of q.
where ̟ denotes a prime in O. The second factor is less than or equal to 1. In the first factor |̟| < 2 1/ε , which is equivalent to d := deg(̟) < 1 ε log 2 log q =: D. Then, Thus, since by [23, Chapter 2], the number a d of primes of degree d satisfies (2.1) Then, using 1 + x e x , we obtain Now, q d /d 2 is increasing with d for q 4 and thus, in this case we have which concludes the proof.
. Then for any ε > 0 and any integer k 2, as claimed.

The circle method over function fields
Recall that k = F q has characteristic > 3 and F ∈ k[x 1 , . . . , x n ] denotes a non-singular homogeneous polynomial of degree 3. Moreover, let X ⊂ P n−1 k be the smooth cubic hypersurface defined by F = 0, and let a, b , and s = t −1 is the prime at infinity. Moreover, where a and b are the corresponding points of a and b in X(k). We remark that any x in the sum has |x| = q d . To simplify notation, we write N a,b (d) = N(d) and Then, by Lemma 2.2, we have By [1, Lemma 4.1], T can be partitioned into a union of intervals centred at rationals and since K is non-archimedean, the intervals do not overlap. Thus, for any Q 1, we have where * denotes a restriction to (a, r) = 1. We shall take Q = 3(d+1) 2 in our work. We now note that S(a/r + θ) is the same as the exponential sum S(a/r + θ) appearing in [1, pg. 690], where b is a and M = t. Define r M = rM/(r, M), for any M, using the notation in [1, (2.4)]. According to [1, Lemma 2.6] we have We note that C ∈ Z. Our strategy is now to go through the remaining arguments in [1, Sections 4 -9] for our particular exponential sums and integrals, paying special attention to the uniformity in the q-aspect. Furthermore, we keep the same notation as in [1,Definition 4.6] for the factorisation of any r ∈ O. Thus, for any j ∈ Z >0 we have r = r j+1 3.1. Exponential Sums. We continue to assume that char (F q ) > 3. Moreover, we note that S r,M,a (c) satisfies the multiplicativity property recorded in [1,Lemma 4.5]. We are interested in the case when M | t.
Lemma 3.2. Let r = uv for coprime u, v ∈ O and t ∤ u. Then there exist non-zero a ′ , a ′′ ∈ k n , depending on a and the residues of u, v modulo t, such that  Proof. This follows directly from [1, Lemma 6.4] on noting that H F = |∆ F | = 1 in our situation.
Let F * ∈ k[x 1 , . . . , x n ] be the dual form of F . Its zero locus parametrises the set of hyperplanes whose intersection with the cubic hypersurface F = 0 produces a singular variety. Moreover, F * is absolutely irreducible and has degree 3 · 2 n−2 . We shall need the following variation of [1,Lemma 6.4] in which the sum is restricted to zeros of F * . Writing z 1 = h + cj with |h| < |c|, we have g(z 1 ) ≡ g(h) + cj∇g(h) mod cd and a 1 ∇g(z 1 ) ≡ a 1 ∇g(h) mod c, since cd | c 2 . Now, g(z 1 ) ≡ 0 mod cd is equivalent to g(h) + cj∇g(h) ≡ 0 mod cd. Thus, g(h) ≡ 0 mod c and we can write g(h) = mc. Thus, m+j∇g(h) ≡ 0 mod d. Moreover, if a 1 ∇g(h) = c+ck, then a 1 g(z 1 ) − c.z 1 ≡ a 1 g(h) − c.h + c 2 (k.j + a 1 h∇g(j)) mod c 2 d, and thus, the sum over z 1 becomes Denote the sum over j by S k,h and estimate it by writing Writing j 1 = j 2 + j 3 and recalling (3.7), we note that for any ε > 0, by [1, Lemma 2.10].
It remains to bound the inner sum. As is [1], let  d 2 ). Thus, we only need to look at the cases when c = ̟ e and d = 1, and c = ̟ e and d = ̟, for any e ∈ Z >0 and any prime ̟. Note that F is non-singular modulo any prime ̟.
Note that this result uses crucially the condition that one of the two fixed points in Theorem 1.1 does not lie on the Hessian of X.

The main term
In this section we investigate the contribution to N(d) in Lemma 3.1 coming from c = 0. Preserving the notation in [1], denote this term by M(d). We will always assume n 10. Thus, Recall that ∇F (t −1 b) = 0 and, in particular, q −2 = |∇F (t −1 b)|. This corresponds to taking ξ = −2 in [1, Section 7.3]. The following result gives a similar bound to that in [1,Lemma 7.4].
Lemma 4.1. For any Y ∈ N and any ε > 0 we have where the implicit constant is independent of q.
Proof. Write r = b 1 b 2 r 3 . Then, by the multiplicativity property in Lemma 5.6, we have , t} and a ′ , a ′′ ∈ k n depend on ord t (r). By [1, Lemma 5.1], where the implicit constant is independent of q. Moreover, where the the implicit constant depends only on n and ε. On noting that for |r| = Y we have |r| −n = q −n Y −n if t ∤ r and |r| −n = Y −n , otherwise, we obtain Then, since # r 3 ∈ O : |r 3 | Y = O( Y 1/3 ) and M 3 ∈ {1, t}, we can bound the above by which concludes the proof.
Then, there exists a constant c, that is independent of q, such that by the same argument as in (2.1). Since exp(z) = 1 + O(|z|), we have q n−1 S = 1 + O(q 2− n 2 ), which concludes the proof.
Thus for n 10, we have where S and J are given by Lemma 4.2 and (4.1), respectively. Note that the error term is satisfactory for Theorem 1.1.
(5.1) We will analyse the contribution to N(d) coming from c = 0 and r, θ such that |r| = Y and |θ| = Θ. Denote this contribution by E(d) = E(d; Y, Θ). This section is similar to [1,Section 7.4] and [1,Section 8], however we need to consider separately the cases when t | r and t ∤ r. Thus, let Moreover, note that Lemma 3.1 imposes a constraint on |c|. More precisely, Moreover, since we must have c = 0 and c ∈ O n , we get the following bound Let S be a set of finite primes to be decided upon in due course but which contains t. Any r ∈ O can be written as From now put b = b ′′ 1 , and d = b ′ 1 r 2 , for simplicity. Also, write S b (c) = S b,1,0 (c). Moreover, take where F * is the dual form of F .
By Lemma 5.6, S b (c) is a multiplicative function of b. Thus, by (5.7) and Lemma 2.4 we have The definition of S and the constraint that (b, S) = 1 imply that and this concludes the proof.
Thus, we will consider separately the case when F * (c) = 0 and the case when F * (c) = 0. Denote the contributions to E(d) coming from c such that F * (c) = 0, respectively F * (c) = 0, by E 1 (d), respectively E 2 (d).
5.1. Treatment of the generic term. Suppose F * (c) = 0. Then, by the first part of Lemma 5.1 we have Then, (5.7) implies that Decompose r 2 as b 2 r 3 . Then, by the multiplicativity property in Lemma 5.6, Moreover, applying Lemma 3.3 with |M 3 | q, Lemma 5.2. For M ′ 2 and b ′ 2 as above and Y ∈ Z, there exists some ε > 0 such that for any c ∈ O n .
Proof. Suppose M ′ 2 = 1 so that S b 2 ,M ′ 2 ,b ′ 2 (c) = S b 2 ,1,0 (c). By [1, (5.3)], together with the fact that in our case |∆ F | = 1, there exists a constant A(n) > 0 depending only on n such that Then, by the multiplicativity property in Lemma 5.6 and by (5.9), It follows from Lemmas 2.4 and 2.5 that this can be bounded by as required.