The following definitions are standard:


An affine variety $V$ in $A^n$ is a complete intersection (c.i.) if its vanishing ideal can be generated by ($n - \dim V$) polynomials in $k[X\_1,\ldots, X\_n]$. The definition can also be made for projective varieties.


$V$ is locally a complete intersection (l.c.i.) if the local ring of each point on $V$ is a c.i. (that is, quotient of a regular local ring by an ideal generated by a regular sequence).


What are examples (preferably affine) of l.c.i. which are not c.i. ? I have never seen such one.



Local complete intersections which are not complete intersections
(To supplement Alberto's example)


If $V$ is projective, then the gap between being locally c.i and c.i is quite big. In particular, any *smooth* $V$ would be locally c.i., but they are not c.i. typically. For instance, take $V$ to be a few points in $\mathbb P^2$ would give simple examples. In higher dimensions, by Grothendick-Lefschetz, if $V$ is smooth, $\dim V\geq 3$, and $V$ is c.i. then $\text{Pic}(V)=\mathbb Z$, so it is a serious restriction.


The affine case is more subtle. Again one can look at smooth varieties. If $V$ is a smooth affine curve and c.i., then the canonical bundle of $V$ is trivial. So it gives the following strategy: start with a *projective* curve $X$ of genus at least $2$, removing some general points to obtain an (still smooth) affine curve with non-trivial canonical bundle.


For more details on the second paragraph, see this [question](https://mathoverflow.net/questions/9751/is-every-smooth-affine-curve-isomorphic-to-a-smooth-affine-plane-curve), especially Bjorn Poonen's comments. This [paper](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-17/issue-3/Complete-intersections/10.1215/kjm/1250522714.full) contains relevant references, and also an example with trivial canonical bundle.


