If $X$ is an affine variety, is $X$ one component of a complete intersection with two?

Question

This is an idle question, but I give the example that motivated me below.

Say $X \subseteq {\mathbb A}^n_k$ is irreducible and $k$ is infinite. Then by picking a regular point of $X$ and picking equations from $X$'s ideal that cut out $T_x X$, we get a scheme containing $X$ as a component.

If we pick those equations generically, can we ensure that that scheme is a complete intersection with at most one extra component beyond $X$?

The example that got me wondering this is where $X = ${$(A,B) : AB = BA$} is the space of pairs of commuting matrices. Then one case of the above construction is $Y = ${$(A,B) : AB-BA$ is diagonal}, which is a reduced complete intersection with two components. I thought this was interesting but now I'm guessing it's the expected behavior.

Answer 1

Perhaps I'm wrong but I thought this was ok by Bertini.

Choose a general hypersurface $H_1$ containing $X$ (ie, choose a general linear combination of the generators of the ideal of $X$, make sure this isn't a pencil). Choose another general hypersurface $H_2$ containing $X$. Repeat this process. Eventually we end up with an intersection of $n - \dim X$ hypersurfaces containing $X$. Call this reducible variety $Y$. This has only one irreducible component besides $X$ by Bertini's theorem (the base locus was $X$ and its not a pencil). Is this what you had in mind?

I assume this must come up in linkage theory (discussed in Eisenbud's book). I believe I've seen this in work of Kawakita and also Ein-Mustata on singularities (see in particular, lci defect ideals).

Answer 2

I will consider the projective closure $X\subset\mathbb{P}^n$. Let say that $X$ is scheme theoretically defined by equations of degree $d_1\geq d_2\geq ... \geq d_m$. Then we can find $f_i \in H^0(X,\mathcal{I}_{X}(d_i))$ for $i = 1,...,c$, where $c = codim_{\mathbb{P}^n}X$ such that $$Y_1\cap...\cap Y_c = X\cup Z,$$ where $Y_i = Z(f_i)$. Furthermore, if $Z$ is non-empty then it is irreducible and intersects $X$ in a divisor.

Basically this comes form liaison theory (http://books.google.it/books/about/Introduction_to_Liaison_Theory_and_Defic.html?id=yiCAwq8XtrkC&redir_esc=y).

You can find the details of the argument here:

• A. Bertram, L. Ein, R. Lazarsfeld, Vanishing theorems, a theorem of Severi and the equations defining projective varieties, J. Amer. Math. Soc. 4 (1991), 587–602.