Elliptic surfaces and K3 surfaces are fascinating objects in algebraic geometry. They combine complex geometry, topology, and arithmetic in unique ways, offering deep insights into the structure of algebraic surfaces.

These surfaces play a crucial role in the classification of algebraic surfaces. Elliptic surfaces provide a bridge between curves and higher-dimensional varieties, while K3 surfaces represent a key class of surfaces with Kodaira dimension zero.

Elliptic Surfaces and Elliptic Curves

Definition and Construction of Elliptic Surfaces

• An elliptic surface is a complex surface that admits a fibration by elliptic curves over a base curve
• The generic fiber of an elliptic surface is an elliptic curve, which is a smooth, projective algebraic curve of genus one with a specified base point
• Elliptic surfaces can be constructed by taking the product of an elliptic curve with another curve and then blowing up the singular points (resolution of singularities)
• The Weierstrass equation can be used to describe elliptic surfaces, with the coefficients being functions on the base curve
  • For example, the Weierstrass equation $y^2 = x^3 + a(t)x + b(t)$ defines an elliptic surface over the base curve with coordinate $t$

Properties and Invariants of Elliptic Surfaces

• The j-invariant of an elliptic surface is a rational function on the base curve that encodes information about the isomorphism class of the fibers
  • Two elliptic curves are isomorphic if and only if they have the same j-invariant
• Elliptic surfaces can have singular fibers, which are fibers that are not smooth elliptic curves
  • Examples of singular fibers include nodal cubic curves, cuspidal cubic curves, and configurations of rational curves
• The Euler characteristic of an elliptic surface can be computed using the topological Euler characteristic of the base curve and the number and types of singular fibers
• Elliptic surfaces with non-constant j-invariant have a positive Euler characteristic, while those with constant j-invariant have Euler characteristic zero

Classification of Elliptic Surfaces

Kodaira Classification of Singular Fibers

• Kodaira classified the possible types of singular fibers that can occur on an elliptic surface
  • Type I$_n$: a cycle of $n$ rational curves (nodal cubic for $n=1$, two rational curves meeting at two points for $n=2$)
  • Type II: a cuspidal cubic curve
  • Type III: two rational curves meeting at a tacnode
  • Type IV: three rational curves meeting at a single point
  • Types I$_n^$, II$^$, III$^$, IV$^$: more complicated configurations of rational curves
• The types of singular fibers that appear on an elliptic surface determine its Euler characteristic and Kodaira dimension

Kodaira Dimension and Euler Characteristic

• The Kodaira dimension of an elliptic surface is determined by its Euler characteristic and the number of singular fibers
  • Kodaira dimension $-\infty$: ruled surfaces (Euler characteristic $< 0$)
  • Kodaira dimension $0$: K3 surfaces, abelian surfaces, Enriques surfaces (Euler characteristic $= 0$)
  • Kodaira dimension $1$: properly elliptic surfaces (Euler characteristic $> 0$)
• Elliptic surfaces with non-constant j-invariant have a positive Euler characteristic, while those with constant j-invariant have Euler characteristic zero
  • For example, a rational elliptic surface (base curve is $\mathbb{P}^1$) with 12 singular fibers of type I$_1$ has Euler characteristic 12

Geometric and Arithmetic Properties of K3 Surfaces

Geometric Properties of K3 Surfaces

• K3 surfaces are complex surfaces with trivial canonical bundle and irregularity zero
  • The canonical bundle is the top exterior power of the cotangent bundle, and its triviality means that K3 surfaces have a unique non-vanishing holomorphic 2-form (up to scaling)
  • Irregularity is the dimension of the first cohomology group with coefficients in the structure sheaf, and its vanishing implies that K3 surfaces have no non-trivial 1-forms
• K3 surfaces are simply connected and have a unique holomorphic 2-form up to scaling
• The second cohomology group of a K3 surface is a 22-dimensional lattice with a specific intersection form
  • The intersection form is even, unimodular, and of signature (3, 19)
  • The lattice is isomorphic to the direct sum of three copies of the hyperbolic plane and two copies of the E$_8$ lattice

Arithmetic Properties of K3 Surfaces

• The Picard group of a K3 surface is a discrete subgroup of the second cohomology group and describes the algebraic curves on the surface
  • The rank of the Picard group (Picard number) is between 0 and 20
  • K3 surfaces with Picard number 20 are called singular K3 surfaces and have interesting arithmetic properties
• K3 surfaces can be realized as double covers of the projective plane branched over a sextic curve or as quartic surfaces in projective 3-space
  • The sextic curve and the quartic surface must satisfy certain conditions to ensure that the double cover or the quartic is a K3 surface
• Elliptic K3 surfaces are K3 surfaces that admit an elliptic fibration, and they have a rich arithmetic structure related to the Mordell-Weil group of sections
  • The Mordell-Weil group is the group of rational points on the generic fiber of the elliptic fibration
  • The rank of the Mordell-Weil group is related to the Picard number of the K3 surface

Moduli Spaces of Elliptic vs K3 Surfaces

Moduli Spaces of Elliptic Surfaces

• The moduli space of elliptic surfaces with a fixed base curve is a complex analytic space that parametrizes isomorphism classes of elliptic surfaces
  • The dimension of the moduli space depends on the genus of the base curve and the number and types of singular fibers
• The moduli space of Jacobian elliptic surfaces, which are elliptic surfaces with a section, is related to the moduli space of stable curves
  • A section of an elliptic surface is a map from the base curve to the surface that is a right inverse to the fibration map
  • The moduli space of stable curves is a compactification of the moduli space of smooth curves that allows certain types of singularities (nodes)

Moduli Spaces of K3 Surfaces

• The moduli space of K3 surfaces is a 20-dimensional quasi-projective variety that can be compactified using the theory of periods
  • The period of a K3 surface is the line integral of the holomorphic 2-form over a basis of the second homology group
  • The period domain is a 20-dimensional complex manifold that parametrizes Hodge structures on the second cohomology of a K3 surface
• The moduli space of polarized K3 surfaces, which are K3 surfaces with a chosen ample line bundle, is a 19-dimensional quasi-projective variety
  • An ample line bundle is a line bundle whose sections define a projective embedding of the surface
  • The degree of the polarization is the self-intersection number of the ample line bundle
• The moduli spaces of elliptic and K3 surfaces have interesting arithmetic properties, such as the existence of dense sets of rational points in certain cases
  • For example, the moduli space of elliptic K3 surfaces with a section has a dense set of rational points corresponding to surfaces with Picard number 20