Limits of Pluri–Tangent Planes to Quartic Surfaces Ciro Ciliberto and Thomas Dedieu
Limits of pluri–tangent planes to quartic surfaces Ciro Ciliberto and Thomas Dedieu Abstract. We describe, for various degenerations S → ∆ of quartic K3 surfaces over the complex unit disk (e.g., to the union of four general planes, and to a general Kummer surface), the limits ∗ as t ∈ ∆ tends to 0 of the Severi varieties Vδ(St), parametrizing irreducible δ-nodal plane sections of St. We give applications of this to (i) the counting of plane nodal curves through base points in special position, (ii) the irreducibility of Severi varieties of a general quartic surface, and (iii) the monodromy of the universal family of rational curves on quartic K3 surfaces. Contents 1 Conventions 4 2 Limit linear systems and limit Severi varieties 4 3 Auxiliary results 10 4 Degeneration to a tetrahedron 11 5 Other degenerations 28 6 Kummer quartic surfaces in P3 30 7 Degeneration to a Kummer surface 33 8 Plane quartics curves through points in special position 41 9 Application to the irreducibility of Severi varieties and to the monodromy action 46 Introduction Our objective in this paper is to study the following: Question A Let f : S → ∆ be a projective family of surfaces of degree d in P3, with S a smooth −1 threefold, and ∆ the complex unit disc (usually called a degeneration of the general St := f (t), for t 6= 0, which is a smooth surface, to the central fibre S0, which is in general supposed to be singular). What are the limits of tangent, bitangent, and tritangent planes to St, for t 6=0, as t tends to 0? Similar questions make sense also for degenerations of plane curves, and we refer to [25, pp.
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