MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 5 down vote accepted

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).

share|cite|improve this answer

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 (

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.
share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.