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

In the last years there have been several new constructions of surfaces of general type with pg = 0, and important progress on their classification. The present paper presents the status of the art on surfaces of general type with pg = 0, and gives an updated list of the existing surfaces, in the case where K 2 = 1;:::; 7. It also focuses on certain important aspects of this classification.

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Surfaces of general type with geometric genus zero: a survey

Shenzhen is one of the most populated city in China, in order to have a deep understanding of the city dynamics, real-time demographic information is very important. In this study, I use mobile phone traffic data with one-hour time resolution and one kilometer spatial resolution to represent population flow. Data during two days are selected, including a Sunday and the following Monday. I firstly create heat maps for each hour in two days and make description analysis combined with statistics indicators. Features and trends are compared among hours and between Monday and Sunday. Then I apply the space time analysis tool to figure out the different types of hot spots trends. After the study, an overall outline of spatial-temporal distribution of population in Shenzhen is made with some interesting facts. However, due to lack of land use and travel survey data, it is still not possible to provide adequate and reasonable explanations for the phenomena.

私の留学体験記 The University of North Carolina at Chapel Hill

The spin moduli spaceSg is the parameter space of theta characteristics (spin structures) on stable curves of genus g. It has two connected components, S g andS + g , depending on the parity of the spin structure. We establish a complete birational classication by Kodaira dimension of the odd componentS g of the spin moduli space. We show thatS g is uniruled for g < 12 and even unirational for g 8. In this range, introducing the concept of cluster for the Mukai variety whose one-dimensional linear sections are general canonical curves of genus g, we construct new birational models ofS g . These we then use to explicitly describe the birational structure of S g . For instance, S 8 is birational to a locally trivial P 7 -bundle over the moduli space of elliptic curves with seven pairs of marked points. For g 12, we prove thatS g is a variety of general type. In genus 12, this requires the construction of a counterexample to the Slope Conjecture on eective divisors on the moduli space of stable curves of genus 12.

/pdf/the-geometry-of-the-moduli-space-of-odd-spin-curves-yriaambgtl.pdf

The geometry of the moduli space of odd spin curves

We prove the irreducibility of universal Severi varieties parametrizing irreducible, reduced, nodal hyperplane sections of primitive K3 surfaces of genus g, with 3 ≤ g≤11, g ≠ 10.

/pdf/on-universal-severi-varieties-of-low-genus-k3-surfaces-4ih87zoxg6.pdf

On universal Severi varieties of low genus K3 surfaces

We investigate the following question: let C be an integral curve contained in a smooth complex algebraic surface X; is it possible to deform C in X into a nodal curve while preserving its geometric genus? We armatively answer it in most cases when X is a Del Pezzo or Hirzebruch surface (this is due to Arbarello and Cornalba, Zariski, and Harris), and in some cases when X is a K3 surface. Partial results are given for all surfaces with numerically trivial canonical class. We also give various examples for which the answer is negative.

/pdf/equigeneric-and-equisingular-families-of-curves-on-surfaces-1c2mf57qec.pdf

Equigeneric and equisingular families of curves on surfaces

Abstract Let C be a smooth projective curve (resp. (S,L){(S,L)} a polarized K⁢3{K3} surface) of genus g⩾11{g\\geqslant 11}, with Clifford index at least 3, considered in its canonical embedding in ℙg-1{\\mathbb{P}^{g-1}} (resp. in its embedding in |L|∨≅ℙg{|L|^{\\vee}\\cong\\mathbb{P}^{g}}). We prove that C (resp. S) is a linear section of an arithmetically Gorenstein normal variety Y in ℙg+r{\\mathbb{P}^{g+r}}, not a cone, with dim⁡(Y)=r+2{\\dim(Y)=r+2} and ωY=𝒪Y⁢(-r){\\omega_{Y}=\\mathcal{O}_{Y}(-r)}, if the cokernel of the Gauss–Wahl map of C (resp. H1⁡(TS⊗L∨){\\operatorname{H}^{1}(T_{S}\\otimes L^{\\vee})}) has dimension larger than or equal to r+1{r+1} (resp. r). This relies on previous work of Wahl and Arbarello–Bruno–Sernesi. We provide various applications.

Wahl maps and extensions of canonical curves and K3K3 surfaces

Self-rational maps of generic algebraic K3 surfaces are conjectured to be trivial. We relate this conjecture to a conjecture concerning the irreducibility of the universal Severi varieties parameterizing nodal curves of given genus and degree lying on some K3 surface. We also establish a number of numerical constraints satisfied by such nontrivial rational maps, that is of topological degree > 1.

/pdf/severi-varieties-and-self-rational-maps-of-k3-surfaces-ag2dn0seou.pdf

Severi varieties and self-rational maps of k3 surfaces

Let X be a Fano manifold such that every rational curve in X has anticanonical degree at least the dimension of X. We prove that X is a projective space or a quadric.

/pdf/numerical-characterisation-of-quadrics-2fm06awnmx.pdf

Thomas Dedieu

Papers

On universal Severi varieties of low genus K3 surfaces

Equigeneric and equisingular families of curves on surfaces

Wahl maps and extensions of canonical curves and K3K3 surfaces

Severi varieties and self-rational maps of k3 surfaces

Numerical characterisation of quadrics