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A Modular Compactification of the Space of Elliptic K3 Surfaces By A Modular Compactification of the Space of Elliptic K3 Surfaces by Adrian Brunyate (Under the Direction of Valery Alexeev) Abstract We describe work towards a compact moduli space of elliptic K3 surfaces with marked divisor given by a small multiple of the sum of rational curves in an ample class of sufficiently large degree. Index words: K3 Moduli, Elliptic surfaces. A Modular Compactification of the Space of Elliptic K3 Surfaces by Adrian Brunyate BS, Towson University, 2006 A Dissertation Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Athens, Georgia 2015 c 2015 Adrian Brunyate All Rights Reserved A Modular Compactification of the Space of Elliptic K3 Surfaces by Adrian Brunyate Approved: Major Professor: Valery Alexeev Pete L. Clark Mitchell Rothstein Committee: Robert Varley Electronic Version Approved: Suzanne Barbour Dean of the Graduate School The University of Georgia August 2015 Acknowledgments I would like to thank my advisor, Valery Alexeev, for introducing me to this material and without whom this work would have been impossible. I would like to thank my family, friends and colleagues for their support, even when I didn't deserve it. iv Contents Acknowledgements iv 1 Introduction 1 I Background 4 2 K3 Surfaces 5 2.1 Introduction and Examples . 5 2.2 Basic Properties . 6 3 Toric Geometry 9 3.1 Affine Toric Varieties . 10 3.2 Non-Affine and Projective Toric Varieties. 11 4 The Minimal Model Program and Moduli of Stable Pairs 16 4.1 Prehistory: Moduli of Pointed Curves . 16 4.2 Singularities in the Minimal Model Program . 18 4.3 The Minimal Model Program and Application to Moduli . 21 4.4 Application to the Moduli of Varieties of Non-General Type . 24 5 Some Lattice Theory 25 v 5.1 Definitions and general theory. 25 5.2 Even Root Lattices . 28 6 Elliptic Surfaces 40 6.1 Weierstrass Fibrations . 45 6.2 The Mordell-Weil Lattice . 49 6.3 Elliptic K3 Surfaces . 50 7 Hodge theory of Kulikov degenerations 52 7.1 Kulikov Degenerations . 53 7.2 Definitions and Common Results . 55 7.3 The Global Torelli Theorem and Type I Degenerations . 59 7.4 Type II . 61 7.5 Type III . 66 8 Compactifications of D=Γ 74 8.1 The Baily-Borel Construction . 75 8.2 The Toroidal Construction . 77 Σ 8.3 Explicit description of F ell ............................ 78 II A Modular Compactification 83 9 KSBA Stable Limits 84 9.1 Description of Limit Pairs . 85 9.2 Singularities of Stable Models . 90 9.3 Components of Stable Pairs . 94 10 Torelli type theorems for components of degenerations and applications. 102 vi 10.1 Torelli Theorems . 102 10.2 Automorphisms of stable pairs . 109 11 Construction of type III Degenerations 113 12 Description of the Conjectural Fan 120 vii Chapter 1 Introduction The problem of finding a natural compact moduli space of (some class of) algebraic K3 sur- faces has a long history. Immediately from the global Torelli theorem of Piatetski-Shapiro and Shafarevich, which expresses the moduli of K3's with some polarization as a quotient of a type IV symmetric domain, we have a natural compactification, the Baily-Borel com- pactification. Unfortunately the resulting space is quite singular, and so the search for compactifications with rich geometric meaning continues. Friedman and Scattone [Fri84] [FS86] discuss partial compactifications from a Hodge theoretic perspective, which agree with Mumford's toroidal construction. Scattone [Sca87] raises the question, as yet unan- swered, as to whether there exist natural toroidal compactifications of a moduli of algebraic K3's. Various authors have used GIT to address the same problem. In particular, Shah [Sha80] produced a geometric compactification of degree 2 K3's as a blowup of a quotient of the space of plane sextics. Also of relevance to the current project, Miranda [Mir89] considered a GIT quotient of the space of Weierstrass equations to obtain a compactification of a moduli space of elliptic K3's. 1 Recently a technology has been developed to produce in a canonical way compactifications of moduli of surfaces of general type, or pairs (X; B) of log general type, introduced by Koll´ar and Shepherd-Baron [KSB88], and Alexeev [Ale94]. It is tempting to use this method to study the K3 case. This was done by Laza [Laz12], who studied the case of degree 2 pairs (X; B) with B arbitrary. If one wishes to remove the element of choice for B, there are several options for canonically associating an ample divisor to a polarized K3. For example, in the degree 2 case one may take the ramification of the involution. This situation is currently being studied by Alexeev and A. Thompson. Alternatively, it was suggested to P study B = Bi, where Bi are the rational curves in the polarization class. We provisionally RC call the corresponding moduli spaces F2d (for rational curves), where 2d is the degree of the polarization. In this thesis I describe work towards continuing this program in the case of (jacobian) elliptic K3 surfaces using the latter approach. We call the resulting space Fell. Beyond it's RC intrinsic interest this lends insight into the (much harder) problem of describing F2d , since RC Fell is exactly the intersection of F2d for all d (though the modular interpretation changes slightly with small d). The main \result" of this work is the conjecture: ν J Conjecture 1.0.1. The normalization Fell of Fell is a toroidal compactification F ell of the period domain Fell corresponding to the fan J described in chapter (12). The main actual results are paraphrased below: Theorem 1.0.2. Let (X; B) be a stable pair parameterized by Fell. Then X is a Weierstrass P24 fibration over a chain of rational curves, and B = i=1 fi + δs, where fi are some fibers and s is the section. The specific pairs that occur are of course enumerated when the theorem is fully stated. This allows a description of each boundary stratum. In particular: 2 ν Theorem 1.0.3. The boundary strata of Fell n Fell are isomorphic to the boundary strata J of F ell. We now briefly discuss the layout of this document. Part I consists of background mate- rial, with the aim being to introduce just enough concepts and notation to understand the second part. While the reader may safely skip to the new results, certain crucial calculations are performed as examples when the appropriate concepts are introduced. The reader is referred back to these when they are used. Also, the level of detail in the exposition varies not only depending on the results needed but also on the technicality of the proofs. While toric geometry (3), elliptic surfaces (6), and lattices (5) are elementary, the Minimal Model Program (4), Mixed Hodge Theory (7), and toroidal compactification (8) are intimidating machines, and it would lead too far afield to even hint at the proofs of many key results. In any case the proofs provided more to provide the flavor of the arguments than to completely develop the results. The most assiduous reader is referred to the references. The second part details new results. First the possible stable pairs are enumerated (9). Next the type III boundary components are parameterized (10). Explicit semistable models are constructed for degenerations and it is shown that all the possible limits described earlier in fact occur (11). Finally the fan J is described, and the isomorphism of boundary strata is shown (12). Finally, a word on the technical approach used. Many statements here have a distinctly \old fashioned" feel, and it is likely that a modern approach using stacks and log geometry would simplify much of this work. The reason for this is two fold. First, the old fashioned approach was easier for me to learn. Since a dissertation is partly a historical artifact of the student's learning process this shapes the exposition enormously. Second, some of the main inspirations for this study predate the development of these tools, and so they need to be integrated along with the primary sources. I hope that readers will be able to shape this discussion to their personal tastes. 3 Part I Background 4 Chapter 2 K3 Surfaces 2.1 Introduction and Examples K3 surfaces were first introduced by Weil in the late 1950's. He chose the name \K3" in honor of Kummer, Kodaira, and K¨ahler,as well as the Himalayan peak. The definition is: Definition 2.1.1. A K3 surface over k is a complete nonsingular algebraic surface X=k with 1 KX = 0 1 2 H (X; OX ) = 0 A polarized K3 surface is a pair (X; L) where X is a K3 surface and L 2 Pic(X) is an ample line bundle. The degree of the surface is L2. In this work, we will assume k = C unless otherwise specified. In the category of complex manifolds notice that the above definition still holds. We recall from the classification of surfaces that the only complete algebraic surfaces with trivial canonical class are abelian surfaces and K3 surfaces, so we can replace \2" in the definition with any other property that rules out abelian surfaces, for example: 5 2' π1(X) trivial. Notice this also rules out Kodaira surfaces (which have odd first Betti number), so in fact serves as an alternate definition of complex analytic K3 surfaces. Because of the fact that they naturally form an analytic family, for algebraic purposes it is often easier to deal with polarized K3's. Example 2.1.2.
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