Sphere Packings, Lattices and Groups

K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

Lectures on K3 Surfaces

Théorie de Hodge et géomětrie algěbrique complexe

Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the 11-dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on Morrison–Nikulin involutions, these are Nikulin involutions which interchange two copies of E
 8(−1) in the Neron Severi group.

Nikulin involutions on K3 surfaces

https://pub.math.leidenuniv.nl/~luijkrmvan/K3Banff/papers/Sarti_symplecticprime_final.pdf

Symplectic automorphisms of prime order on K3 surfaces

Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice U 3 ⊕ E 8(−1)2 depends only on the group but not on the K3 surface For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces

/pdf/elliptic-fibrations-and-symplectic-automorphisms-on-k3-2ypsnm775k.pdf

Elliptic Fibrations and Symplectic Automorphisms on K3 Surfaces

The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a carefull analysis of the divisors contained in the Picard lattice we study their projective models, giving necessary and sufficient conditions to have an even set. Moreover we investigate their relation with K3 surfaces with a Nikulin involution.

/pdf/projective-models-of-k3-surfaces-with-an-even-set-3yehzptsi5.pdf

Projective models of k3 surfaces with an even set

Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice $U^3\oplus E_8(-1)^2$ depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces.

Elliptic fibrations and symplectic automorphisms on K3 surfaces

Hesse claimed in [7] (and later also in [8]) that an irreducible projective hypersurface in ℙn defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved in [6] that this is true forn≤3 and constructed counterexamples for everyn≥4. Gordan and Noether and Franchetta gave classification of hypersurfaces in ℙ4 with vanishing hessian and which are not cones, see [6, 5]. Here we translate in geometric terms Gordan and Noether approach, providing direct geometrical proofs of these results.

https://link.springer.com/content/pdf/10.1007/BF03191214.pdf

Alice Garbagnati

Papers

Symplectic automorphisms of prime order on K3 surfaces

Elliptic Fibrations and Symplectic Automorphisms on K3 Surfaces

Projective models of k3 surfaces with an even set

Elliptic fibrations and symplectic automorphisms on K3 surfaces

A Geometrical approach to Gordan–Noether’s and Franchetta’s contributions to a question posed by Hesse