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Let $X$ be a topological space and denote by $F_n(X)$ the following subspace: $$F_n(X):=\{(x_1,\cdots ,x_n)\in X^n: x_i\neq x_j \forall i\neq j\}.$$ Note that, we are not considering the quotient of $F_n$ by $\Sigma_n$. Is there a decent analogue of this construction for general schemes?

Basically, one could consider the naive translation in algebraic geometry, but from my experience, this is usually not the right way to go.

Thanks in advance. Cheers.

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    $\begingroup$ What is the problem with applying this definition directly to schemes? $\endgroup$
    – Lazzaro Campeotti
    Commented Oct 11, 2016 at 19:47
  • $\begingroup$ Well, the first problem is, what should points be? prime ideals? geometric points? $\endgroup$
    – I.P
    Commented Oct 11, 2016 at 19:54
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    $\begingroup$ If $X$ is a separated scheme over $S$, you can take $F_2(X)$ to be the complement $X\times_S X-\Delta$. Similarly for any $n$, take the complement of the "big" diagonal in $X^n = X\times_S X\ldots$ This looks pretty reasonable. $\endgroup$
    – Donu Arapura
    Commented Oct 11, 2016 at 20:00
  • $\begingroup$ That seems to be the type of definition that would work best. However, what if $\geq 2$? what if $X$ is not separated? Is there an established analogue of this for schemes? $\endgroup$
    – I.P
    Commented Oct 11, 2016 at 20:02
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    $\begingroup$ I had in mind the complement of the biggest thing that could be called a diagonal, i.e. the locus where two coordinates are equal. More precisely, $x_i=x_j$ corresponds to a closed immersion $X^{n-1}\to X^n$. Take the union of all of these, and then form the complement. $\endgroup$
    – Donu Arapura
    Commented Oct 11, 2016 at 20:33

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