Local complete intersections which are not complete intersections - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T23:48:13Z http://mathoverflow.net/feeds/question/27197 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27197/local-complete-intersections-which-are-not-complete-intersections Local complete intersections which are not complete intersections Adam K 2010-06-05T21:36:06Z 2011-04-27T21:18:22Z <p>The following definitions are standard:</p> <p>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.</p> <p>$V$ 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).</p> <p>What are examples (preferably affine) of l.c.i. which are not c.i. ? I have never seen such one.</p> http://mathoverflow.net/questions/27197/local-complete-intersections-which-are-not-complete-intersections/27199#27199 Answer by Alberto García-Raboso for Local complete intersections which are not complete intersections Alberto García-Raboso 2010-06-05T21:38:18Z 2010-06-05T21:38:18Z <p>The first example is the <a href="http://en.wikipedia.org/wiki/Twisted_cubic" rel="nofollow">twisted cubic</a> in $\mathbb{P}^3$.</p> http://mathoverflow.net/questions/27197/local-complete-intersections-which-are-not-complete-intersections/27203#27203 Answer by Hailong Dao for Local complete intersections which are not complete intersections Hailong Dao 2010-06-05T22:26:10Z 2010-06-05T23:09:07Z <p>(To supplement Alberto's example)</p> <p>If $V$ is projective, then the gap between being locally c.i and c.i is quite big. In particular, any <em>smooth</em> $V$ 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. </p> <p>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 <em>projective</em> curve $X$ of genus at least $2$, removing some general points to obtain an (still smooth) affine curve with non-trivial canonical bundle. </p> <p>For more details on the second paragraph, see this <a href="http://mathoverflow.net/questions/9751/is-every-smooth-affine-curve-isomorphic-to-a-smooth-affine-plane-curve" rel="nofollow">question</a>, especially Bjorn Poonen's comments. This <a href="http://projecteuclid.org/DPubS?verb=Display&amp;version=1.0&amp;service=UI&amp;handle=euclid.kjm/1250522714&amp;page=record" rel="nofollow">paper</a> contains relevant references, and also an example with trivial canonical bundle. </p> http://mathoverflow.net/questions/27197/local-complete-intersections-which-are-not-complete-intersections/27210#27210 Answer by Matthew Morrow for Local complete intersections which are not complete intersections Matthew Morrow 2010-06-05T23:28:14Z 2010-06-05T23:58:14Z <p>EDIT: This is wrong. I haven't deleted it in order that the subsequent comments make sense.</p> <p>You will never <strong>see</strong> an example, for the following reason: given a local complete intersection $V$ inside $\mathbb{A}_k^n$, you can always find a global complete intersection $W$ inside $\mathbb{A}_k^n$ such that the reduced varieties associated to $V$ and $W$ are the same.</p> <p>Proof:</p> <p>Suppose $I$ is an ideal of $k[X_1,\dots,X_n]$ such that the variety $V(I)$ is a local complete intersection. This forces all the local rings of $V(I)$ to be Cohen-Macaulay, hence equidimensional. So the irreducible components of $V(I)$ all have the same codimension in $\mathbb{A}_k^n$; lets call this codimension $r$.</p> <p>Since $k[X_1,\dots,X_n]$ is Cohen-Macaulay, the height of $I$ (which is $r$) is the same as its depth, meaning that $I$ contains a regular sequence $f_1,\dots,f_r$ of length $r$. By considering heights we see that the minimal primes over the ideal $J=\langle f_1,\dots,f_r\rangle$ are the same as the minimal primes over $I$. Therefore $J$ and $I$ have the same radical, which implies the claim (with $W=V(J)$). QED</p> <p>So if you are trying to draw counterexamples, you have to worry about whether that line on the paper has nilpotent elements in the structure sheaf...</p> http://mathoverflow.net/questions/27197/local-complete-intersections-which-are-not-complete-intersections/27230#27230 Answer by Adam K for Local complete intersections which are not complete intersections Adam K 2010-06-06T09:15:07Z 2010-06-06T09:15:07Z <p>From Hailong's answer, I suppose it is possible to make simpler examples as follows: take $V$ a smooth affine variety which is not equidimensional (so clearly it is l.c.i but not c.i). For instance, $V$ is the union of the plane $z = 0$ and the line $z=1, x=y$ in $\mathbb A^3$. $V$ is smooth (it can be proven that $I(V) = (zx-zy, z^2-z)$).</p> <p>The disadvantage of this construction is $V$ must be reducible.</p> <p>Please correct me if I'm wrong.</p>