K. Matsuki | Searcharxiv

arXiv2000

A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension

The recent two proofs for the (weak) factorization theorem for birational maps, one by Włodarczyk and the other by Abramovich-Karu-Matsuki-Włodarczyk rely on the results of Morelli. The former uses the process for $π$-desingularization (the most subtle part of Morelli's combinatorial algorithm), while the latter uses the strong factorization of toric (toroidal) birational maps directly. This paper provides a coherent account of Morelli's work (and its toroidal extension) clarifying some discrepancies in the original argument.

math.AG↗

arXiv1994

Mumford-Thaddeus Principle on the Moduli Space of Vector Bundles on an Algebraic Surface

We study the behavior of the Gieseker space of semistable torsion-free sheaves of rank r and fixed c_1, c_2 on a non-singular projective surface as the polarization varies. It is shown that the ample cone admits a locally finite chamber structure, and that passing a wall adjacent to a pair of chambers has the effect of modifying the moduli space by a (finite) sequence of flips of the type studied by Thaddeus. The key steps are a modification of Simpson's method and the introduction of a "rationally twisted" moduli space. The result is more general but less explicit than the recent work of Ellingsrud-Goettsche (alg-geom/9410005) and Friedman-Qin (alg-geom/9410007).

alg-geom↗