There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

This book can form the basis of a second course in algebraic geometry. As motivation, it takes concrete questions from enumerative geometry and intersection theory, and provides intuition and technique, so that the student develops the ability to solve geometric problems. The authors explain key ideas, including rational equivalence, Chow rings, Schubert calculus and Chern classes, and readers will appreciate the abundant examples, many provided as exercises with solutions available online. Intersection is concerned with the enumeration of solutions of systems of polynomial equations in several variables. It has been an active area of mathematics since the work of Leibniz. Chasles' nineteenth-century calculation that there are 3264 smooth conic plane curves tangent to five given general conics was an important landmark, and was the inspiration behind the title of this book. Such computations were motivation for Poincare's development of topology, and for many subsequent theories, so that intersection theory is now a central topic of modern mathematics.

3264 and All That

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Revetements Etales Et Groupe Fondamental Sga 1

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

In this paper we present the classification of non-symplectic automorphisms of prime order on K3 surfaces, i.e. we describe the topological structure of their fixed locus and determine their invariant lattice in cohomology. We provide new results for automorphisms of order 5 and 7 and alternative proofs for higher orders. Moreover, for any prime p, we identify the irreducible components of the moduli space of K3 surfaces with a non-symplectic automorphism of order p.

K3 surfaces with non-symplectic automorphisms of prime order

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

Symplectic automorphisms of prime order on K3 surfaces

In this note we present the classification of non-symplectic automorphisms of prime order on K3 surfaces, i.e.we describe the topological structure of their fixed locus and determine the invariant lattice in cohomology. We provide new results for automorphisms of order 5 and 7 and alternative proofs for higher orders. Moreover, for any prime p, we identify the irreducible components of the moduli space of K3 surfaces with a non-symplectic automorphism of order p.

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the eleven 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.

Alessandra Sarti

Papers

Nikulin involutions on K3 surfaces

K3 surfaces with non-symplectic automorphisms of prime order

Symplectic automorphisms of prime order on K3 surfaces

K3 surfaces with non-symplectic automorphisms of prime order

Nikulin involutions on K3 surfaces