Do subsets of generators of a toric ideal generate a toric ideal?

Question

Given a toric ideal, say $J$, in a polynomial ring $k[x_1,...,x_n]$ we can find a finite generating set for $J$. Is it possible, perhaps under additional assumptions on the structure of $J$, to give a finite minimal generating set for $J$ such that every subset of generators also generates a toric ideal.

If not, are there any known counterexamples in the general case?

If yes, could you provide a reference? Does it generalize to lattice ideals?

Motivation: For particularly chosen, generating sets of toric ideals there exist subsets that also generate a toric ideal. For e.g. the 2xn determinantal ideal is toric. A generating set is given by the 2x2 minors of the defining matrix, say $M$. Then the ideal generated by those minors which correspond to a subset of the columns of the $M$ is also toric.

Answer 1

I believe this is a counterexample. Toric ideals are prime by definition (assuming that I am remembering correctly). Then I think that the ideal of the twisted cubic $C\subseteq \mathbb P^3$ ought to do it.

$J=\langle xz-y^2, xw-yz,wy-z^2\rangle$.

Any two of the three generators will intersect in the union of $C$ and a line. For instance, the vanishing of $I=\langle xz-y^2, xw-yz\rangle$ is $C\cup L$ where $L=V(x,y)$. So $I$ can't be prime.

Answer 2

What is going on can be explained completely in terms of exponent vectors. To do so we can map each generator $x^u -x^v$ of $J$ to the exponent vector $u-v \in \mathbb{Z}^n$. Conversely, for each integer vector $s\in \mathbb{Z}^n$ we can form a binomial $x^{s^+} - x^{s^-}$ where $s^\pm_i = \max(\pm s_i, 0)$ are the positive and negative part of $s$, such that $s = s^+ - s^-$. Now, take the integer vector images of a minimal generating set of $J$ and let $L_J$ be the sublattice of $\mathbb{Z}^n$ that they generate. This lattice is independent of the chosen generators and we have the following basic fact:

Prop. Let $S \subset L$ be any finite set of vectors spanning $L$ as a lattice. Then $J = \langle x^{s^+} - x^{s^-} : s\in S\rangle : \left(\prod_i x_i\right)^\infty$.

Typically a generating set of $L$ will be much smaller than any image of a minimal generating set of $J$. (Google for "Markov basis" to find tons of examples from algebraic statistics.) The proposition says that $J$ can be reconstructed from $L$, but also that whenever you remove a generator from $J$ and $L$ does not change, then the resulting ideal can not be prime (it has some extra components that are contained in the union of the coordinate planes). Dan's example is of this form.

Therefore the only chance that you may have, is when $J$ is, in fact, a complete intersection. This case is treated in the paper "Affine semigroup rings that are complete intersections" by Fischer, Morris, and Shapiro Proc. AMS 125 (11), 1997. It contains a combinatorial characterization complete intersection semigroups in Theorem 3.1 which may be useful for finding a counterexample in that case too.