In "All toric l.c.i.-singularities admit projective crepant resolutions", the authors say (second page)

Families of Gorenstein non-l.c.i. toric singularities that have such special full resolutions seem to exist only rarely.

I would like to think they would not say this if there were no examples of Gorenstein non-l.c.i. toric singularities.

Just to emphasize what everyone else is saying, being l.c.i. is an intrinsic and local condition on a ring. So the natural question is to ask about an affine toric variety, since the question for a projective toric varitey just comes down to checking in every coordinate chart. And l.c.i. implies Gorenstein, which not all toric varieties are, so you at least want to include the condition that your toric variety be Gorenstein.

I think I have an example. Let $D$ be the lattice $\{(i,j,k,\ell) \in \mathbb{Z}^4 : i+j+k+\ell \equiv 0 \mod 2 \}$. Let $C$ be the cone $\{ i,\ j,\ k,\ \ell \geq 0 \}$ in $D \otimes \mathbb{R}$. Our toric singularity will be $S:=k[C \cap D]$. Note that we can think of this as the subring of $k[w,x,y,z]$ consisting of even degree polynomials. So $S$ is generated by $w^2$, $wx$, ..., $z^2$ -- ten monomials in all.

Normal semi-group rings are always Cohen-Macaulay, and the dualizing module is $\mathrm{Span}_k (C^{\circ} \cap D)$ where $C^{\circ}$ is the interior of $C$. In other words, the dualizing module consists of the $k$-span of the even degree monomials where each variable appears with even degree. This module is clearly generated by $wxyz$, so the dualizing module is free of rank $1$, and we see that $S$ is Gorenstein.

Now, let's check that $S$ is not l.c.i. We see $S$ as a quotient of the polynomial ring in $10$ variables, coming from the generators $w^2$, $wx$ etc. Let $R$ be the polynomial ring in $10$ variables. I'll write $[w^2]$ and so forth for the generators of $R$. Let $I$ be the kernel of $S \to R$; we want to show that $I$ is NOT generated by any $6$ elements. Let $\mathcal{M}$ be the maximal ideal of $R$. Since $I$ is graded, it is enough to check that $I/\mathcal{M}I$ is NOT $6$ dimensional.

Well, there are already $6$ degree $2$ elements of $I$ which are the $S_4$ symmetries of $[w^2] [x^2] = [wx]^2$. In addition, we have two more relations $[wx] [yz] = [wy][xz] = [wz][xy]$. These $8$ elements are linearly independent in $I/\mathcal{M} I$, so we see that $S$ is not an l.c.i.

In "All toric l.c.i.-singularities admit projective crepant resolutions", the authors say (second page)

Families of Gorenstein non-l.c.i. toric singularities that have such special full resolutions seem to exist only rarely.

I would like to think they would not say this if there were no examples of Gorenstein non-l.c.i. toric singularities.

Just to emphasize what everyone else is saying, being l.c.i. is an intrinsic and local condition on a ring. So the natural question is to ask about an affine toric variety, since the question for a projective toric varitey just comes down to checking in every coordinate chart. And l.c.i. implies Gorenstein, which not all toric varieties are, so you at least want to include the condition that your toric variety be Gorenstein.

I think I have an example of a Gorenstein toric non-l.c.i. Let $D$ be the lattice $\{(i,j,k,\ell) \in \mathbb{Z}^4 : i+j+k+\ell \equiv 0 \mod 2 \}$. Let $C$ be the cone $\{ i,\ j,\ k,\ \ell \geq 0 \}$ in $D \otimes \mathbb{R}$. Our toric singularity will be $S:=k[C \cap D]$. Note that we can think of this as the subring of $k[w,x,y,z]$ consisting of even degree polynomials. So $S$ is generated by $w^2$, $wx$, ..., $z^2$ -- ten monomials in all.

Normal semi-group rings are always Cohen-Macaulay, and the dualizing module is $\mathrm{Span}_k (C^{\circ} \cap D)$ where $C^{\circ}$ is the interior of $C$. In other words, the dualizing module consists of the $k$-span of the even degree monomials where each variable appears with even degree. This module is clearly generated by $wxyz$, so the dualizing module is free of rank $1$, and we see that $S$ is Gorenstein.

Now, let's check that $S$ is not l.c.i. We see $S$ as a quotient of the polynomial ring in $10$ variables, coming from the generators $w^2$, $wx$ etc. Let $R$ be the polynomial ring in $10$ variables. I'll write $[w^2]$ and so forth for the generators of $R$. Let $I$ be the kernel of $S \to R$; we want to show that $I$ is NOT generated by any $6$ elements. Let $\mathcal{M}$ be the maximal ideal of $R$. Since $I$ is graded, it is enough to check that $I/\mathcal{M}I$ is NOT $6$ dimensional.

Well, there are already $6$ degree $2$ elements of $I$ which are the $S_4$ symmetries of $[w^2] [x^2] = [wx]^2$. In addition, we have two more relations $[wx] [yz] = [wy][xz] = [wz][xy]$. These $8$ elements are linearly independent in $I/\mathcal{M} I$, so we see that $S$ is not an l.c.i.

In "All toric l.c.i.-singularities admit projective crepant resolutions", the authors say (second page)

Families of Gorenstein non-l.c.i. toric singularities that have such special full resolutions seem to exist only rarely.

I would like to think they would not say this if there were no examples of Gorenstein non-l.c.i. toric singularities.

Just to emphasize what everyone else is saying, being l.c.i. is an intrinsic and local condition on a ring. So the natural question is to ask about an affine toric variety, since the question for a projective toric varitey just comes down to checking in every coordinate chart. And l.c.i. implies Gorenstein, which not all toric varieties are, so you at least want to include the condition that your toric variety be Gorenstein.

In "All toric l.c.i.-singularities admit projective crepant resolutions", the authors say (second page)

Families of Gorenstein non-l.c.i. toric singularities that have such special full resolutions seem to exist only rarely.

I would like to think they would not say this if there were no examples of Gorenstein non-l.c.i. toric singularities.

Just to emphasize what everyone else is saying, being l.c.i. is an intrinsic and local condition on a ring. So the natural question is to ask about an affine toric variety, since the question for a projective toric varitey just comes down to checking in every coordinate chart. And l.c.i. implies Gorenstein, which not all toric varieties are, so you at least want to include the condition that your toric variety be Gorenstein.

I think I have an example. Let $D$ be the lattice $\{(i,j,k,\ell) \in \mathbb{Z}^4 : i+j+k+\ell \equiv 0 \mod 2 \}$. Let $C$ be the cone $\{ i,\ j,\ k,\ \ell \geq 0 \}$ in $D \otimes \mathbb{R}$. Our toric singularity will be $S:=k[C \cap D]$. Note that we can think of this as the subring of $k[w,x,y,z]$ consisting of even degree polynomials. So $S$ is generated by $w^2$, $wx$, ..., $z^2$ -- ten monomials in all.

Normal semi-group rings are always Cohen-Macaulay, and the dualizing module is $\mathrm{Span}_k (C^{\circ} \cap D)$ where $C^{\circ}$ is the interior of $C$. In other words, the dualizing module consists of the $k$-span of the even degree monomials where each variable appears with even degree. This module is clearly generated by $wxyz$, so the dualizing module is free of rank $1$, and we see that $S$ is Gorenstein.

Now, let's check that $S$ is not l.c.i. We see $S$ as a quotient of the polynomial ring in $10$ variables, coming from the generators $w^2$, $wx$ etc. Let $R$ be the polynomial ring in $10$ variables. I'll write $[w^2]$ and so forth for the generators of $R$. Let $I$ be the kernel of $S \to R$; we want to show that $I$ is NOT generated by any $6$ elements. Let $\mathcal{M}$ be the maximal ideal of $R$. Since $I$ is graded, it is enough to check that $I/\mathcal{M}I$ is NOT $6$ dimensional.

Well, there are already $6$ degree $2$ elements of $I$ which are the $S_4$ symmetries of $[w^2] [x^2] = [wx]^2$. In addition, we have two more relations $[wx] [yz] = [wy][xz] = [wz][xy]$. These $8$ elements are linearly independent in $I/\mathcal{M} I$, so we see that $S$ is not an l.c.i.