3-Webs generated by confocal conics and circles

Abstract

We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them.

Webs of rational curves on real surfaces and a classification of real weak del Pezzo surfaces

“…Suppose that Y → X is the smooth model of a weak del Pezzo surface X ⊂ P n with canonical class k. Either one of the following eight cases holds (we use Notation 2 and Table 1). (a) k 2 = 9, Y ∼ = P 2 and G(X) = {− 1 3 k}.…”

“…Similarly, if Y is the blowup of P 2 in four complex conjugate points, then without loss of generality 2e 0 − e 1 − e 2 − e 3 − e 4 ∈ F R (X). If Y is the blowup of P 1 × P 1 in two complex conjugate points, then the pullback of a curve of bidegree (1,1), that passes through these points, has class c ∈ F (X) such that σ * (c) = c so that…”

“…The classification problem for hexagonal webs of circles is known as the Blaschke-Bol problem [2, § 3, Aufgabe 1, p. 31]. The general classification might be a very difficult problem, but some recent progress has been made [1,23,31].…”

“…If X ⊂ P n is a real surface such that G(X) contains two disjoint vertices, then there exists a real birational map P 1 × P 1 X of bidegree (d, d), where d is the degree of a simple curve on X. Moreover, if a real birational map P 1 × P 1 X is of bidegree (a, b), then a, b d.…”

“…Proof of Corollary 4. By assumption, X ⊂ P n contains at least two conics through each point, and thus we know from [29] (alternatively see [20,Corollary 2]), that X is either a ruled surface of degree at most 3 or a weak del Pezzo surface of canonical degree 3 k 2 9 with h ∈ {− 1 2 k, − 1 3 k, − 2 3 k, −k} the class of hyperplane sections. The linear projection of a hexagonal web is again hexagonal and thus we only have to classify linear normalizations X N ⊂ P m .…”