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Let Y be an affine noetherian scheme, $Z = V_+(F_1, \ldots, F_r)$ a closed subscheme of $\mathbb{P}^d_Y$ that is flat over Y. Let $y_0 \in Y$ be a point such that $Z_{y_0}$ is a complete intersection in $\mathbb{P}^d_{k(y_0)}$. Set $r = dim Z_{y_0}$. I am trying to show that this implies that there is an open neighborhood V of Y such that for all $y \in V$we have that $Z_y$ is a complete intersection in $\mathbb{P}^d_{k(y)}$ of dimension $r$ for every $ y \in V$. I am having no luck here, however. I have tried the following:

  1. Look at the case $r=2$ and show it there. Here still no luck, and the methods I thought of was quite ugly. One was to try to consider $(F_2)/rad(F_1)$ as some sort of quasicoherent sheaf and show something regarding the support. It didn't work however.
  2. Trying to just invert coefficients in polynomials in $F_i$. However, I couldn't show that inverting certain coefficients (those coefficients of $F_i$ not vanishing on $y_0$) gave a local complete intersection.

Any hints or solutions are welcome!

user101036
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1 Answers1

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This is only a comment, not a full answer. I hope it will help you a bit on your way.

Let $Y$ be an affine noetherian scheme, and $y_0$ a regular point of $Y$ such that codim $y_0 =1$. Let $Z\to Y$ be a flat morphism of schemes such that $Z_{y_0}\to $ Spec $k(y_0)$ is a complete intersection. Then, we can use Proposition 2.1.12 in Olivier Benoist's thesis; see http://www-irma.u-strasbg.fr/~benoist/articles/Thesefinale.pdf to see that $Z\to Y$ is a complete intersection over the local ring $\mathcal O_{Y,y_0}$ of $y_0$.

In fact, Benoist proves that, since $T= $ Spec $\mathcal O_{Y,y_0}$ is a discrete valuation ring (by the regularity and codimension assumptions on $y_0$), the scheme $Z_T:= Z\times_Y T$ is a complete intersection over $T$, and moreover that the equations for $Z_{y_0}\to $ Spec $k(y_0)$ lift to equations for $Z_T\to T$. (Here we use that there is a canonical morphism $T\to Y$.) Actually, we probably don't need the second part of the result of Benoist to answer your question, but it's good to note it.

Ariyan Javanpeykar
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