Xxxx’x algorithm Sample Clauses

Xxxx’x algorithm. Trees and Canonicality‌ A singular elliptic fibration with a section can be realized in terms of a Weierstrass model y2 = x3 + fxw4 + gw6 , (5.1.9) where [w, x, y] are homogenous coordinates in P[1,2,3]. Let z be a local coordinate on the base of the fibration and let z = 0 be a component of the discriminant of the Weierstrass 1As will be explained in the next section in codimension 1 any vanishing order in b0 can be shifted or ‘lopped’ away, so that this always has a zeroeth order term. model ∆ = 4f 3 + 27g2. We will assume throughout that the divisor z = 0 in the base is smooth. The possible singular fibers in codimension one in the base were classified by Kodaira and N´xxxx [84–86]2. For a given singular Weierstrass model in P[1,2,3] Xxxx’x algorithm [1,88,89] allows a systematic determination of the singular fibers in codimension one in the base of the fibration. The algorithm is based upon successively determining the conditions for the vanishing of the discriminant in the coordinate z in the base. The coordinate ring in a sufficiently small neighborhood on the divisor z = 0 in the base is a unique factorization domain (UFD) [147]. Xxxx’x algorithm proceeds then by solving the conditions ∆ = 0 order by order in an expansion in z over a UFD. In the process the Weierstrass form can be brought into the so-called Xxxx form y2 + b1xy + b3y = x3 + b2x2 + b4x + b6 , (5.1.10) where the coefficients bi are sections of suitable line bundles, and have an expansion in powers of z that characterize the singular fibers. We will refer to a Xxxx form as canonical, if it is characterized solely by the vanishing orders of the coefficients bi. As was shown in [1], most Weierstrass forms in P[1,2,3] can be locally put into (canonical) Xxxx forms, albeit there exist outliner cases, which cannot be reached without allowing for divisions, in which case only generalized Xxxx forms can be achieved locally. These non-canononical forms are not specified solely by a vanishing order of the coefficients, but require non- trivially relation among the coeffficients bi, which cannot be removed by well-defined coordinate changes like shifts.
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