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A variety $X$ is a locally set theoretic complete intersection in a nonsingular variety $Y \supseteq X$ if each point in $X$ has a neighborhood $U$ in $Y$ such that $X \cap U$ is set theoretically the set of zeroes of precisely $k$ regular functions on $U$, where $k$ is the codimension of $X$ in $Y$.

Question: Is every irreducible variety a locally set theoretic complete intersection (in an appropriate ambient space)?

It is clear that every nonsingular variety is a locally set theoretic complete intersection. My guess was that it is not true in general; however I can not come up with a counterexample.

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    $\begingroup$ Presumably, if there were a subvariety $X \subset \mathbb{P}^{n-1}$ which was not a set theoretic complete intersection, then the cone $CX \subset \mathbb{A}^n$ would not be a local stci ? According to mathoverflow.net/questions/190938 , it is expected that the rational quartic (image of $\mathbb{P}^1 \to \mathbb{P}^3$ under $(x:y) \mapsto (x^4 : x^3 y : x y^3 : y^4)$) is not an scti in characteristic zero, but this is still open. On a quick google, I couldn't find ANY subvariety of $\mathbb{P}^n$ or $\mathbb{A}^n$ which is known NOT to be an scti. $\endgroup$
    – David E Speyer
    Commented Mar 19 at 21:27
  • $\begingroup$ If $X$ is locally a stci in a(ny) smooth variety, say over $\mathbf{C}$, then the shifted constant sheaf $\mathbf{Q}[\mathrm{dim} X]$ is perverse. There are plenty of $X$ for which this fails. $\endgroup$
    – Satan's Minion
    Commented Mar 20 at 6:32

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At least if $k=\mathbb{C}$, the article Certain cones are not set-theoretic complete intersections, R. R. Simha , Arch. Math. (Basel)27(1976), no.2, 169–171 should give an example. The example is the affine cone in $\mathbb{C}^6$ of the Segre embedding of $\mathbb{P}^1\times\mathbb{P}^2$ in $\mathbb{P}^5$.

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  • $\begingroup$ This is great - thanks! $\endgroup$
    – pinaki
    Commented Mar 20 at 17:21

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