Cissoid

Abstract: Let `1;:::` l be l lines in P 2 such that no three lines meet in a point. Let X(l) be the set of points fi \ `j j1 � i < jlgP 2 . We call X(l) a star configuration. We describe all pairs (d;l) such that the generic degree d curve in P 2 contains a X(l).

Abstract: The conchoid of a plane curve C is constructed using a fixed circle B in the affine plane. We generalize the classical definition so that we obtain a conchoid from any pair of curves B and C in the projective plane. We present two definitions, one purely algebraic through resultants and a more geometric one using an incidence correspondence in P 2 × P 2. We prove, among other things, that the conchoid of a generic curve of fixed degree is irreducible, we determine its singularities and give a formula for its degree and genus. In the final section we return to the classical case: for any given curve C we give a criterion for its conchoid to be irreducible and we give a procedure to determine when a curve is the conchoid of another.

Abstract: A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1 - o(1)) n(2). We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.) An important ingredient of our proofs is the following statement. Let S be a family of n open curves in R-2, so that each curve is the graph of a continuous real function defined on R, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is Omega(nt root logt/log log t).