2022/11/13 by Deng, Jiahe, Kretschmer, Andreas
#13D02 #13P10 #14N10 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2211.07020
We study algebraic and homological properties of the ideal of submaximal minors of a sparse generic symmetric matrix. This ideal is generated by all (n-1)-minors of a symmetric n × n matrix whose entries in the upper triangle are distinct variables or zeros and the zeros are only allowed at off-diagonal places. The surviving off-diagonal entries are encoded as a simple graph G with n vertices. We prove that the minimal free resolution of this ideal is obtained from the case without any zeros via a simple pruning procedure, extending methods of Boocher. This allows us to compute all graded Betti numbers in terms of n and a single invariant of G. Moreover, it turns out that these ideals are always radical and have Cohen--Macaulay quotients if and only if G is either connected or has no edges at all. The key input are some new Gröbner basis results with respect to non-diagonal term orders associated to G.