Algebraic and Complex Geometry by Anne Frühbis-Krüger Remke Nanne Kloosterman & Matthias Schütt

Author:Anne Frühbis-Krüger, Remke Nanne Kloosterman & Matthias Schütt
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Corollary 2.

In the general fibre of the generic P 2 bundle structure of , the Severi variety of 1–nodal (resp. 2–nodal) irreducible curves is an irreducible curve of degree 10 (resp. the union of 16 distinct points).

Proof.

This follows from the fact that the above mentioned Severi varieties are respectively the dual curve of a plane quartic as in Proposition 12, and the set of ordinary double points of . One computes the degrees using Plücker formulae. □

5.6 The Linear Systems

Here we study some geometric properties of the linear systems appearing in the third step of Sect. 5.3.

Consider a triangle L 1, L 2, L 3 in P 2, with vertices a 1, a 2, a 3, where a 1 is opposite to L 1, etc. Consider the linear system of cubics through a 1, a 2, a 3 and tangent there to L 3, L 1, L 2 respectively. By Proposition 9, there is a birational transformation of T k to the plane (see Fig. 5) mapping to . We consider the rational map (or simply ϕ) determined by the linear system , or, alternatively, the rational map, with the same image T (up to projective transformations), determined by the planar linear system . The usual notation is {1, …, 4} = { i, j, s, k}.

Proposition 13.

The map ϕ: T k → T ⊂ P 3 is a birational morphism, and T is a cubic surface with three double points of type A 2 as its only singularities. The minimal resolution of T is the blow–down of T k contracting the (−1)–curves . This cubic contains exactly three lines, each of them containing two of the double points.

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