Four counterexamples in combinatorial algebraic geometry

We present counterexamples to four conjectures which appeared in the Literature in commutative algebra and algebraic geometry. The four questions to be studied are largely unrelated, and yet our answers are connected by a common thread: they are combinatorial in nature, involving monomial ideals and binomial ideals, and they were found by exhaustive computer search using the symbolic algebra systems Maple and Macaulay 2. In Section 1 we answer Chandler's question [4, Question 1] whether the Castelnuovo-Mumford regularity of a homogeneous polynomial ideal I satisfies the inequality reg(I-r) less than or equal to r . reg(I). We present a characteristic-free counterexample generated by only eight monomials; this improves an earlier example by Terai [5, Remark 3]. In Section 2 we settle a conjecture published two decades ago by Briancon and Iarrobino [2, p. 544], by showing that the most singular point on the Hilbert scheme of points need not be the monomial ideal with most generators. In Section 3 we construct a smooth projectively normal curve which is defined by quadrics but is not Koszul: this solves a problem posed by Butler [3, Problem 6.5] and Polishchuk [16, p. 123]. Section 4 disproves an overly optimistic conjecture of mine [23, Example 13.17] about the Grobner bases of a certain toric 4-fold. Each of the four counterexamples is displayed in the user language of Macaulay 2; see [11]. We encourage the readers to try out these lines of code and to enjoy their own explorations in combinatorial algebraic geometry. Naturally, our results raise many more questions than they answer, and several new open problems will be stated in this article.