Blowing Up Toric Varieties with Multidimensional Continued Fractions

Because toric varieties are built up from convex geometry, there is a natural connection to be made with triangle partition maps, which are multidimensional continued fraction algorithms. Our motivation to explore this connection is showing that to resolve the curve yp=xq we follow a path of blowups given by the continued fraction expansion of pq. Dividing the triangle according to the Triangle Map turns out to be equivalent to blowing up an axis in C3. We apply these blowups to resolving singularities of curves. We also discuss these blowups and blowdowns in terms of attracting or repelling curves toward or away from curves defined by a quadratic irrational or a pair of cubic irrationals with a periodic triangle sequence.

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