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Laumon’s resolution of Drinfeld’s compactification is small. (English) Zbl 0910.14026
Let $$C$$ be a smooth projective curve of genus 0. Let $${\mathcal B}$$ be the variety of complete flags in an $$n$$-dimensional vector space $$V$$. Given an $$(n-1)$$-tuple $$\alpha$$ of positive integers one can consider the space $${\mathcal Q}_\alpha$$ of algebraic maps of degree $$\alpha$$ from $$C$$ to $${\mathcal B}$$. This space has drawn much attention recently in connection with quantum cohomology [see e.g. A. B. Givental, Int. Math. Res. Not. 1996, No. 13, 613-663 (1996; Zbl 0881.55006) and M. Kontsevich in: The moduli space of curves, Proc. Conf., Texel Island 1994, Prog. Math. 129, 335-368 (1995; Zbl 0885.14028)]. The space $${\mathcal Q}_\alpha$$ is smooth but not compact. The problem of compactification of $${\mathcal Q}_\alpha$$ proved very important. One compactification $${\mathcal Q}^K_\alpha$$ was constructed by M. Kontsevich (loc. cit.) (the space of stable maps). Another compactification $${\mathcal Q}^L_\alpha$$ (the space of quasiflags), was constructed by G. Laumon in: Automorphic forms, Shimura varieties, and $$L$$-functions, Vol. I, Proc. Conf., Ann. Arbor 1988, Perspect. Math. 10, 227-281 (1990; Zbl 0773.11032). However, historically the first and most economical compactification $${\mathcal Q}^D_\alpha$$ (the space of quasimaps) was constructed by Drinfeld (early 80-s, unpublished). The latter compactification is singular, while the former ones are smooth. Drinfeld has conjectured that the natural map $$\pi:{\mathcal Q}^L_\alpha \to {\mathcal Q}^D_\alpha$$ is a small resolution of singularities.
In the present note we prove this conjecture after the necessary recollections. The arguments in the proof are rather similar to those of G. Laumon (loc. cit., 3.3.2). In fact, the proof gives some additional information about the fibers of $$\pi$$. – In conclusion, let us mention that the Drinfeld compactifications are defined for the space of maps into flag manifolds of an arbitrary semisimple group, and it would be very interesting to construct their small resolutions.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 32J05 Compactification of analytic spaces
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