Algebraic versus symplectic… blowups

It’s hard to miss the many parallels between symplectic toric geometry and algebraic toric geometry, which makes learning one having already experienced much of the other quite peculiar. I’m going to write a few posts outlining some of the similarities (and differences) between toric geometry in the algebraic and symplectic contexts, starting with blowups. Blowups provide, for me at least, a good insight into the level of functoriality present between the combinatorics and geometry at hand. For instance, on the algebraic side they arise from the simplest of maps – the identity – on the level of lattices, but are sufficiently rich to provide such resolutions as the Jung-Hirzebruch resolution discussed in previous posts (which is the minimal resolution of isolated surface quotient singularities) and describe G\text{-Hilb}. It has been remarked to me that the combinatorial presentation of blowups on the symplectic side is more appealing, which I agree with to a point: one is more overtly replacing a torus fixed point with a projective space. Though of course there one can’t so easily (or at all?) treat the nonsmooth case. Swings and roundabouts.

delz

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