We define a family of homogeneous ideals with large projective dimension and regularity relative to the number of generators and their common degree. This family subsumes and improves upon constructions given in [Cav04] and [McC]. In particular, we describe a family of three-generated homogeneous ideals in arbitrary characteristic whose projective dimension grows asymptotically as √ d √ d−1 .

Let T (Z m × Z n ) denote the maximal number of points that can be placed on an m × n discrete torus with "no three in a line," meaning no three in a coset of a cyclic subgroup of Z m × Z n . By proving upper bounds and providing explicit constructions, for distinct primes p and q, we show thatVia Gröbner bases, we compute T (Z m × Z n ) for 2 ≤ m ≤ 7 and 2 ≤ n ≤ 19.

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We give different short proofs for a result proved by C. Mueller in [9]: Over an algebraically closed field pairs of n × n matrices whose product is symmetric form an irreducible, reduced, and complete intersection variety of dimension (3n 2 + n)/2. Our work is connected to the work of Brennan, Pinto, and Vasconcelos in [2].

We study a generalization of the Canonical Element Conjecture. In particular we show that given a nonregular local ring (A, m) and an i > 0, there exist finitely generated A-modules M such that the canonical map from Ext i A (M/mM, Syz i (M/mM)) to H i m (M, Syz i (M/mM)) is nonzero. Moreover, we show that even when M has an infinite projective dimension and i > dim(A), studying these maps sheds light on the Canonical Element Conjecture.