A bound on the number of points of a plane curve

A conjecture is formulated for an upper bound on the number of points in PG(2, q) of a plane curve without linear components, defined over GF(q). We prove a new bound which is half way from the known bound to the conjectured one. The conjecture is true for curves of low or high degree, or with rational singularity. Let C be a plane curve of degree n, defined over GF(q), without (rational) linear components. Let Mq denote the number of points of the projective plane PG(2, q) satisfying the equation of C, counted without multiplicity. In this short note we discuss upper bounds on Mq. For the number of rational points the well-known bound is Nq ≤ q+1+ (n− 1)(n− 2) √ q if C is absolutely irreducible ((Hasse-)Weil [9]); and we also have the combinatorial Mq ≤ (n− 1)q+ n (Barlotti [2], Thas [7]). Thas proved Mq ≤ (n − 1)q + n − 2 (if n > 2) and there were other improvements on this bound but under strong additional conditions only. [1, 4] and [8] use the assumption n|q while [6] either gives a small improvement on the bound or uses an assumption n ≪ q. In fact these (more general) results give bounds for the size of a (k, n)-arc, a point set intersecting every line in ≤ n points; the set of points in PG(2, q) on C is obviously a special (k, n)-arc. Here the following conjecture is made. Conjecture 1 A curve of degree n defined over GF(q), without linear components, has always Mq ≤ (n − 1)q + 1 points in PG(2, q). If true then for n = 1, 2, √ q+1, q−1 it would be sharp as the curves X2−Y Z,X √ +Y √ + Z √ q+1 and αX + βY q−1 − (α+ β)Z (where α, β, α+ β 6= 0) show. Note that Lunelli and Sce conjectured the similar bound for (k, n)-arcs (and that conjecture was false). The conjecture is true if there exists a line skew to the curve and (q, n) = 1, see Blokhuis [3]; also if there exists a line with 1 rational point of C, see below; or if n ≥ q + 2. If C has a rational singular point P then each line through P contains ≤ n − 2 further points of C so Mq ≤ (n − 2)(q + 1) + 1. (So from now on Mq = Nq can be assumed.) We also remark that it is enough to prove the conjecture for absolutely irreducible curves; then for general C it can be proved by induction: let C split to the absolutely irreducible components C1 ∪ C2 ∪ ... ∪ Ck with degrees n1, ..., nk; if each Ci had ≤ (ni − 1)q + 1 points then in total C would have ≤ ∑ki=1(niq− q+1) = nq− k(q− 1) < nq− q+1 points. So at least one of them, Cj say, has more than njq − q + 1 points, so more than n2j (if nj < q, which can be supposed); so by [5], Lemma 2.24(i), Cj can be defined over GF(q) and then the induction hypothesis finishes the proof. As a corollary we immediately see that if n ≤ √q+1 then q+1+(n−1)(n−2)√q ≤ (n−1)q+1 proves the conjecture by Weil’s bound. Note that by the reasoning above, if C cannot be defined over GF(q) and n 6= q, q + 1 then the bound in the conjecture is true. ∗Research is supported by OTKA F-043772, T-043758 grants