Hi
Let $V$ be an affine or projective variety. Recall that $V$ is a local complete intersection (l.c.i) if all its local rings are complete intersection. Also recall that $V$ is a complete intersection (c.i) if $I(V)$ is generated by a regular sequence of length $codim(V)$. 


I do not know a single example of a l.c.i variety which is not c.i and NOT smooth. Examples will be greatly appreciated.



Example of locally complete intersection varieties which are not smooth and not complete intersection
If $X\subset \mathbb P^N$ is an l.c.i projective subvariety that linearly spans $\mathbb P^N$, and if $p\in\mathbb P^n\setminus X$ is a point s.t. the projection of $X$ from $p$ into $\mathbb P^{N-1}$ is an isomorphism onto its image $X'$, then $X'$ will not be a complete intersection (since the linear system of its hyperplane section is not complete).


To obtain such an example, suppose that $Y\subset\mathbb P^N$ is a smooth surface not lying in a hyperplane and that $N$ is big enough ($N\ge6$ will suffice). If characteristic is zero, there exists a quadric $Q\subset\mathbb P^N$ that is tangent to $Y$ at exactly one point $y\in Y$. Now if $X=Y\cap Q$ and $P$ is a generic point of $\mathbb P^N$ (in particular, $p$ should not lie in the tangent space $T\_yY$), then the projection $\pi\_p\colon X\to \mathbb P^{N\_1}$ is an isomorphism onto ints image, and $\pi\_p(X)\subset\mathbb P^{N-1}$ is an l.c.i that is not a c.i.


