Rees Algebras of Closed Determinantal Facet Ideals

Abstract

Using SAGBI basis techniques, we find Gröbner bases for the presentation ideals of the Rees algebra and special fiber ring of a closed determinantal facet ideal. In particular, we show that closed determinantal facet ideals are of fiber type and their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains, and have rational singularities.

Powers of binomial edge ideals with quadratic Gröbner bases

“…G . Since J G = P W (G)∈Ass J G P W (G), we show g ∈ (P W (G)) 2 , for all W with the property that P W (G) ∈ Ass J G . Then it follows that g ∈ J…”

“…Case 1. If W ∩ [6] is equal to ∅ or {1} or {4} or {6}, then g = x 5 y 4 (x 6 y 2 − x 2 y 6 )(x 3 y 1 − x 1 y 3 ) + x 3 y 6 (x 4 y 2 − x 2 y 4 )(x 1 y 5 − x 5 y 1 ) ∈ (P W (G)) 2 .…”

“…W ∩ [6] = {3}. Then g = x 1 x 3 y 3 y 4 (x 2 y 6 − x 6 y 2 ) + x 3 x 4 y 1 y 2 (x 5 y 6 − x 6 y 5 ) + x 1 x 3 y 4 y 6 (x 5 y 2 − x 2 y 5 ) + x 3 x 5 y 2 y 4 (x 1 y 6 − x 6 y 1 ) + x 3 x 4 y 2 y 5 (x 5 y 6 − x 6 y 5 ) ∈ P W (G) 2 .…”

“…W ∩ [6] = {5}. Then g = x 1 x 3 y 5 y 6 (x 4 y 2 − x 2 y 4 ) + x 5 x 6 y 2 y 4 (x 3 y 1 − x 1 y 3 ) + x 2 x 5 y 3 y 6 (x 1 y 4 − x 4 y 1 ) + x 4 x 5 y 1 y 6 (x 2 y 3 − x 3 y 2 ) ∈ (P W (G)) 2 .…”

Rees algebras of ideals of star configurations

“…On the other hand, only a few classes of ideals have well-understood free resolutions, and usually a free resolution for I does not provide information on the free resolutions of its powers I m . Nevertheless, the problem becomes manageable if one imposes algebraic conditions on a presentation matrix of I (see for instance [29,25,21]), especially when one can exploit methods from algebraic combinatorics (see for instance [31,18,10,8,12,15,1]).…”