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Grauert and Manin showed that a non-isotrivial family of compact complex hyperbolic curves has finitely many sections. We consider a generic moving enough family of high enough degree hypersurfaces in a complex projective space. We show the existence of a strict closed subset of its total space that contains the image of all its sections.

Families of quintic surfaces and curves

Edmond E. Griffin (1985)

Compositio Mathematica

Families of smooth curves on surface singularities and wedges

Gérard Gonzalez-Sprinberg, Monique Lejeune-Jalabert (1997)

Annales Polonici Mathematici

Following the study of the arc structure of singularities, initiated by J. Nash, we give criteria for the existence of smooth curves on a surface singularity (S,O) and of smooth branches of its generic hypersurface section. The main applications are the following: the existence of a natural partition of the set of smooth curves on (S,O) into families, a description of each of them by means of chains of infinitely near points and their associated maximal cycle and the existence of smooth curves on...

Families of supersingular abelian surfaces

Toshiyuki Katsura, Frans Oort (1987)

Compositio Mathematica

Fano bundles of rank 2 on surfaces

Michał Szurek, Jarosław A. Wisniewski (1990)

Compositio Mathematica

Fano fourfolds of index greater than one.

P.M.H. Wilson (1987)

Journal für die reine und angewandte Mathematik

Fano hypersurfaces in weighted projective 4-spaces.

Johnson, Jennifer M., Kollár, János (2001)

Experimental Mathematics

Fano manifolds and quadric bundles.

Jaroslaw A. Wisniewski (1993)

Mathematische Zeitschrift

Fano manifolds and vector bundles.

Thomas. Peternell, Michal Szurek (1992)

Mathematische Annalen

Fano manifolds of degree ten and EPW sextics

Atanas Iliev, Laurent Manivel (2011)

Annales scientifiques de l'École Normale Supérieure

O’Grady showed that certain special sextics in $ℙ^{5}$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

Fano Schemes of Linear Spaces on Hypersurfaces.

Adrian Langer (1997)

Manuscripta mathematica

Fermatquadrupel.

Werner Meyer, Wolfram Neutsch (1981)

Mathematische Annalen

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