Singular Elliptic Fibrations and their Resolutions Sample Clauses

Singular Elliptic Fibrations and their Resolutions. ‌ As we shall see, 7-branes in F-theory and the non-abelian gauge theories they host are realized as loci where the complex structure of the elliptic fiber, and with it the string coupling constant, degenerates. These are given by the singular loci of the elliptic fi- bration. In this section, we develop the notion of singular loci of algebraic varieties and techniques to study them. These will allow us in the next section to completely classify singularities of elliptic fibrations, as well as construct normal forms for them. In section 3.2, we will then map this classification into a classification of non-abelian gauge groups of their associated F-theory models. A singular point of an algebraic variety V is a point at which the tangent space of V degenerates. Specifically for a hypersurface F (x, y, z, . . . ) = 0, the singular points are those where all partial derivatives of F vanish. For an elliptic curve described by a Weierstraß equation PW , the tangent space xxxxxxx- ations are given when dPW = 0. First, consider the coordinate patch of P[1,2,3] where w does not vanish, and use the scaling relation to set w = 1. The remaining coordinates are x and y, and they are not related by a scaling relation anymore. For brevity, we will call this the w = 1 patch, and define the x = 1 patch and the y = 1 patch analogously. Here, PW = dPW = 0 reads ∂yPW = 2y = 0 PW = x3 + fx + g = (x − a)(x − b)(x − c) = 0 ∂xPW = (x − a)(x − b) + (x − a)(x − c) + (x − b)(x − c) = 0 ,
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