Some general results on plane curves with total inflection points

Abstract

In this paper we study plane curves of degree d with e total inflection points, for nonzero natural numbers d and e.Mathematics Subject Classification (2000). 14H10, 14H50, 14N20.Keywords. Plane curve, total inflection point, nodal curve. Introduction.Let e and d be nonzero natural numbers and denote by V d,e the set of elements ((L 1 , P 1 ), . . . , (L e , P e )) with L 1 , . . . , L e lines in the complex projective plane P 2 , P i a point on the line L i for each i, P i ∈ L j for each i = j and such that there exists a plane curve C of degree d with contact order d with L i at P i for each i. So P 1 , . . . , P e are total inflection points of C.In [5], the case d = 4 (i.e. quartic curves) has been studied intensively. The main tool used in that thesis is the so-called λ-invariant, which is nothing else than a cross ratio of four points. In [3], the cases e = 1, 2 have been handled and also the cases e = 3, 4 for some special configurations of the lines L i and the points P i .In Section 1 of this paper, we will first prove a general result (Proposition 1.3). Note that part (a) of this Proposition is already known (see [5, Chapter II, Lemma 2.15] or [3, Theorem A]). In Section 2, we prove a generalization of the main result of a paper of E. Ballico (see [2]). In Section 3, we introduce the notion of expected dimension for a component of V d,e .In a following paper of the authors, the established techniques of this paper will be used to describe all the components of V d,e in case e is equal to 3 or 4, together with their images in the moduli spaces. Both authors are partially supported by the Fund of Scientific Research -Flanders (G.0318.06). Definition and a general result.Definition 1.1. Let P 2 be the complex projective plane and P 2 be the incidence relation in (P 2 ) * × P 2 , thus P 2 = {(L, P )|P ∈ L}.

Star points on smooth hypersurfaces

“…We are going to generalize Proposition 1.3 in [3] to the case of star points. First we introduce some terminology.…”

“…In case e = 3, we find f = 3d, hence dim(V d, 3 ) 15. In case V d, 3 would have a component of dimension exactly 15, it would be an orbit under the action of Aut(P 3 ).…”

“…In case V d, 3 would have a component of dimension exactly 15, it would be an orbit under the action of Aut(P 3 ). Combining this description with Theorem 2.10, the space of hypersurfaces with three star points has dimension at least 15 + d 3 .…”

“…So we have either t d = 1 or t d−1 = 1. In these cases, for general A j ∈ C and 3 corresponding to the root of unity t = 1 by V t and denote the order of t in C by θ(t). If we fix L ∈ V t , the values of the numbers A j are fixed (up to a scalar), but g 012 can vary, so the dimension of P d (L) equals d 3 .…”

“…As a starting point in [3], we examined possible configurations of points and lines that can appear as such total inflection points and the associated tangent lines. In this paper, we generalize this point of view to higher-dimensional varieties.…”