Cotas para o número máximo de retas duas a duas disjuntas em uma família S

Abstract

Let r(S) be the maximum number of pairwise disjunct lines that a non-singular surface S ⊂ P3 contains and rd = max {r(S) | deegre(S) = d}. Ensure that r(S) = 6 for all non-singular cubic surface S, therefore r3 = 6. For d = 4, r4 = 16, it was showed by the Russian mathematician Viacheslav Nikulin in [9]. We quote that Rojas-Santos in [7], obtained that r(F) = 16 if F is the Schur’s quartic. At the moment rd is unknown for d ≥ 5. In this work we aim to present bounds for the maximum number of two-by-two disjunct straight lines in the family S whose members are the deegre d non-singular surfaces Sd ⊂ P3 definided by φ(x0,x1)−φ(x2,x3) being φ(u,v) = uv(ud−2−vd−2) and d ≥ 5. In fact, for d odd we show that r(Sd) = d(d−2) + 4, however Boiss´ere-Sarti proved that r(Sd) ≥ d(d−2) + 4 when d is odd and d ≥ 7 in [3]. For the even case, we obtain d(d−2) + 4 ≤ r(Sd) ≤ d(d−2) + d2 2 if d 6= 6 and r(S6) = 48. Considering the bound rd ≤ d(d−2) for all d ≥ 4 given by the Japanese mathematician Miyaoka in [8], we conclude as soon as r6 = 48.