I am studying first order deformations and a natural question arises.

Situation: Let $X_1$ be a scheme. $\pi: X_1 \to {\rm Spec}~ k[t]/(t^2)$ is a flat morphism of finite type, where $k$ is an algebraically closed field. $X_0$ is the central fiber, i.e., $X_1 \times_{{\rm Spec}~ k} {\rm Spec~} k[t]/(t^2)$. The subscript $0$ will always denote the central fiber of the corresponding object. Assume $X_0$ is regular. $U_1\subset X_1$ is an open subset and $Z_1\subset U_1$ is an effective Cartier divisor.

Question: Is the scheme-theoretic closure of $Z_1$ in $X_1$ an effective Cartier divisor?

My Observations:

  1. An effective Cartier divisor $W_1 \subset X_1$ is flat over ${\rm Spec~}k[t]/(t^2)$.
  2. As in 1, $W_0$ is an effective Cartier divisor in $X_0$.
  3. If $W_1$ is a subscheme of $X_1$, flat over ${\rm Spec~}k[t]/(t^2)$, assume $W_0$ is an effective Cartier divisor, then $W_1$ is an effective Cartier divisor. [Lemma 10.93.1, Stacks Project, Tag 00MD]
  4. $\bar {Z_0}$ need not equal $\bar {Z_1}\cap X_0$. Example: $X_1=\mathbb P^2_{{\rm Spec~}k[t]/(t^2)}$, $U_1={\rm Spec~} k[x,y,t]/(t^2)$ and $Z_1={\rm Spec~} k[x,y,t]/(t^2,x+ty^2)$.
  5. [This is wrong. See the answer.]If for every $x\in X_0$, there is a neighborhood $U$ such that the class group $Cl(U)=0$, then the answer to the above question is positive. This is because locally the deformation is trivial, so we may assume that $X_1={\rm Spec~}\left( A\oplus tA\right)$ where $A$ is some $k-$alegebra which is an UFD. Then we can compute the scheme theoretic closure easily.

[This is wrong. See the answer.]By observation 5, if there would be a counterexample, $X_0$ cannot be as simple as affine planes.


1 Answer 1


Sorry, I do not understand what you say in 5. What if A = k[x, y] and X_0 = Spec(A) and U_0 = D(x) and the effective Cartier divisor is given by the zero locus of y + t/x ? Then (y + t/x)(y - t/x) = y^2 and the scheme theoretic closure of Z_1 is given by an ideal containing both y^2 and xy + t which cannot be principal.

There are results in the literature concerning this question when one takes out something of sufficiently high codimension and X_0 is sufficiently nice. Look in Grothendieck's Cohomologie locale des faisceaux coh´erents et th´eoremes de Lefschetz locaux et globaux (SGA 2) for example.

  • $\begingroup$ Yes you are definitely right! I made a mistake here. Thank you! $\endgroup$
    – Yang Zhou
    Jan 2, 2014 at 0:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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