A D 5 elliptic fibration is a fibration whose generic fiber is modeled by the complete intersection of two quadric surfaces in P 3 . They provide simple examples of elliptic fibrations admitting a rich spectrum of singular fibers (not all on the list of Kodaira) without introducing singularities in the total space of the fibration, thus avoiding a discussion of their resolutions. We study systematically the fiber geometry of such fibrations using Segre symbols and compute several topological invariants.We present for the first time Sen's (orientifold) limits for D 5 elliptic fibrations. These orientifolds limits describe different weak coupling limits of F-theory to type IIB string theory giving a system of three brane-image-brane pairs in presence of a Z 2 orientifold. The orientifold theory is mathematically described by the double cover the base of the elliptic fibration. Such orientifold theories are characterized by a transition from a semi-stable singular fiber to an unstable one. In this paper, we describe the first example of a weak coupling limit in F-theory characterized by a transition to a non-Kodaira (and non-ADE) fiber. Inspired by string dualities, we obtain non-trivial topological relations connecting the elliptic fibration and the different loci that appear in its weak coupling limit. Mathematically, these are very surprising relations which relate the total Chern class of the D 5 elliptic fibration and those of different loci that naturally appear in the weak coupling limit. We work over bases of arbitrary dimension and our results are independent of any Calabi-Yau hypothesis.

A D 5 elliptic fibration is a fibration whose generic fiber is modeled by the complete intersection of two quadric surfaces in P 3 . They provide simple examples of elliptic fibrations admitting a rich spectrum of singular fibers (not all on the list of Kodaira) without introducing singularities in the total space of the fibration and therefore avoiding a discussion of their resolutions. We study systematically the fiber geometry of such fibrations using Segre symbols and compute several topological invariants.We present for the first time Sen's (orientifold) limits for D 5 elliptic fibrations. These orientifolds limit describe different weak coupling limits of F-theory to type IIB string theory giving a system of three brane-image-brane pairs in presence of a Z 2 orientifold. The orientifold theory is mathematically described by the double cover the base of the elliptic fibration. Such orientifold theories are characterized by a transition from a semi-stable singular fiber to an unstable one. In this paper, we describe the first example of a weak coupling limit in F-theory characterized by a transition to a non-Kodaira (and non-ADE) fiber. Inspired by string dualities, we obtain non-trivial topological relations connecting the elliptic fibration and the different loci that appear in its weak coupling limit. Mathematically, these are very surprising relations relating the total Chern class of the D 5 elliptic fibration and those of different loci that naturally appear in the weak coupling limit. We work in arbitrary dimension and are result don't assume the Calabi-Yau condition.

Abstract. We present a formula for computing proper pushforwards of classes in the Chow ring of a projective bundle under the projection π : P(E ) → B, for B a non-singular compact complex algebraic variety of any dimension. Our formula readily produces generalizations of formulas derived by Sethi,Vafa, and Witten to compute the Euler characteristic of elliptically fibered Calabi-Yau fourfolds used for F-theory compactifications of string vacua. The utility of such a formula is illustrated through applications, such as the ability to compute the Chern numbers of any non-singular complete intersection in such a projective bundle in terms of the Chern class of a line bundle on B.

We derive a formula for the Milnor class of scheme-theoretic global complete intersections (with arbitrary singularities) in a smooth variety in terms of the Segre class of its singular scheme. In codimension one the formula recovers a formula of Aluffi for the Milnor class of a hypersurface.

We provide a stochastic extension of the Baez–Fritz–Leinster characterization of the Shannon information loss associated with a measure-preserving function. This recovers the conditional entropy and a closely related information-theoretic measure that we call conditional information loss. Although not functorial, these information measures are semi-functorial, a concept we introduce that is definable in any Markov category. We also introduce the notion of an entropic Bayes’ rule for information measures, and we provide a characterization of conditional entropy in terms of this rule.