In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1.
(Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perhaps without a chosen origin.)
The surface and the base curve are assumed to be non-singular (complex manifolds or regular schemes, depending on the context).
Both elliptic and singular fibers are important in string theory, especially in F-theory.
The following table lists the possible fibers of a minimal elliptic fibration.
("Minimal" means roughly one that cannot be factored through a "smaller" one; precisely, the singular fibers should contain no smooth rational curves with self-intersection number −1.)
Geometric arguments show that the intersection matrix of the components of the fiber must be negative semidefinite, connected, symmetric, and have no diagonal entries equal to −1 (by minimality).
The intersection matrix determines the fiber type with three exceptions: The monodromy around each singular fiber is a well-defined conjugacy class in the group SL(2,Z) of 2 × 2 integer matrices with determinant 1.
Representatives for these conjugacy classes associated to singular fibers are given by:[1] For singular fibers of type II, III, IV, I0*, IV*, III*, or II*, the monodromy has finite order in SL(2,Z).
This reflects the fact that an elliptic fibration has potential good reduction at such a fiber.
For singular fibers, this group structure on the smooth locus is described in the table, assuming for convenience that the base field is the complex numbers.
(For a singular fiber with intersection matrix given by an affine Dynkin diagram
, for an integer mi at least 2 and a divisor Di whose coefficients have greatest common divisor equal to 1, and L is some line bundle on the smooth curve S. If S is projective (or equivalently, compact), then the degree of L is determined by the holomorphic Euler characteristics of X and S: deg(L) = χ(X,OX) − 2χ(S,OS).
The canonical bundle formula implies that KX is Q-linearly equivalent to the pullback of some Q-divisor on S; it is essential here that the elliptic surface X → S is minimal.
Building on work of Kenji Ueno, Takao Fujita (1986) gave a useful variant of the canonical bundle formula, showing how KX depends on the variation of the smooth fibers.
The discriminant divisor in Fujita's formula is defined by where c(p) is the log canonical threshold
This is an explicit rational number between 0 and 1, depending on the type of singular fiber.
The canonical bundle formula (in Fujita's form) has been generalized by Yujiro Kawamata and others to families of Calabi–Yau varieties of any dimension.