Two Characteristic Numbers for Smooth Plane Curves of Any Degree

We use a sequence of blow-ups over the projective space parametrizing plane curves of degree d to obtain some enumerative results concerning smooth plane curves of arbitrary degree. For d = 4 , this gives a first modem verification of results of H. G. Zeuthen. O. INTRODUCTION The kth 'characteristic number' of the d(dt3) -dimensional family of smooth plane curves of degree d, denoted Nd(k) in the following, is the number of such curves which are tangent to k lines and contain d(dt3) k points in general position in the plane. Elementary considerations and Bezout's theorem (see §1 below) show that Nd(k) = (2d 2)k for k < 2d 1. In this paper we compute the next two cases as a closed form in terms of the degree d . Our result is Nd(2d 1) = (2d 2)2d-l 2d3d(d l)(d2 d + 2), Nd(2d) = (2d 2)2d 2d4d(d 1)(8d4 21d3 + 19d2 20d + 32). Also, for d = 4 we obtain the next characteristic number N4 (9) = 9,840,040. The characteristic numbers of a family are its basic enumerative information; the problem of computing them for families of plane curves has received quite some attention in the recent past. For the family of smooth plane curves of degree d, the modem literature lists the numbers N2(k), N3(k) for smooth conics and cubics [F, A, KS]; for d = 4, the numbers N4 (7) = 279,600, N4 (8) = 1,668,096 and N4(9) = 9,840,040 computed here verify classic results of H. G. Zeuthen's [Z] (in which-among many others-all the characteristic numbers N4(k) for smooth plane quartics are obtained). For degree ~ 5, the results of this paper seem to be new (we know of recent work of Leendert van Gastel on this problem, from a different viewpoint). Our approach is in the spirit of the computation of the characteristic numbers for smooth plane cubics in [A]. Let jp'N be the projective space parametrizing plane curves of degree d . Call 'point-condition' the hyperplane in jp'N formed by the curves C E jp'N which contain a given point, and 'line-condition' the hypersurface (of degree 2d 2) consisting of the curves C E jp'N which are Received by the editors July 21, 1989 and, in revised form, March 8, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 14NIO; Secondary 14C17.