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Singularities of certain ladder determinantal varieties. (English) Zbl 0842.13008
A noetherian local ring $$R$$ of prime characteristic is said to be $$F$$-rational if every parameter ideal is tighly closed. In an earlier paper, the second author [“$$F$$-rational rings have rational singularities” (preprint)] showed that $$F$$-rational type implies rational singularities. A. Conca and J. Herzog [“Determinantal rings of one-sided ladders are $$F$$-rational” (preprint)] used this result to prove that one-sided ladder determinantal ideals have rational singularities. This can be deduced also from works of Mulay and Ramanathan on Schubert varieties. Since the classical determinantal varieties have rational singularities, one may ask whether this holds for the larger class of ladder determinantal varieties. The main result of this paper confirms this question for complete intersection ladder determinantal varieties by showing that all ideals of their coordinate rings are tightly closed.

##### MSC:
 13C40 Linkage, complete intersections and determinantal ideals 14B05 Singularities in algebraic geometry 14M12 Determinantal varieties 13B22 Integral closure of commutative rings and ideals
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