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Assume that $X$ is a smooth 3-fold and let $C\subseteq X$ a curve with a unique singular point of multiplicity $2$. Does there exist a smooth surface $S$ inside $X$ which contain $C$ ?

Clearly if the multiplicity of $C$ was at least 3 then it would be very easy to find counter-examples. On the other hand, if the multiplicity is 2, it seems that at least infinitesimally it is possible to find such a surface. In case the answer is negative, is it true at least locally (e.g. in an analytic neighbourhood of p)?

Finally any smooth curve is locally complete intersection. What about the curve $C$ above?

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    $\begingroup$ Once you know that the tangent space of $C$ at the singular point has dimension $2$ (see Hailong Dao's answer and Jason Starr's comment), if $X$ is quasi-projective over a perfect infinite field, then $C$ is contained in a smooth hypersurface of $X$, see Kleiman-Altman: ''Bertini theorems for hypersurfaces containing a subscheme'', Proposition (6). Over a finite field, Poonen : ''Bertini theorems over finite fields'', Theorem 1.3 should do the trick. $\endgroup$
    – Qing Liu
    Commented Jul 19, 2011 at 23:45
  • $\begingroup$ Note that if the base field is imperfect, then even the local statement fails (but OK if replace smooth with regular). $\endgroup$
    – Qing Liu
    Commented Jul 20, 2011 at 8:24

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Here is a quick proof that any complete local Cohan-Macaulay ring of dimension $1$ and multiplicity $2$ is a hypersurface, so the answer to your last question is always yes.

Call such ring $R$ with maximal ideal $m$. Since $e(R)=2$ and $R$ is CM, there is a regular element $x\in m$ such that $$length(R/xR) = e(R)=2$$ (see Bruns-Herzog, chapter 4).

EDIT: as Graham pointed out below, technically for such $x$ to exist one needs $R/m$ to be infinite. But one can enlarge the field without affecting the conclusion that $R$ is a hypersurface.

Now the left hand side is $length(m/xR) +1$, so $length(m/xR)=1$, so $m/xR$ is one-generated. It follows that $m$ is $2$-generated. Thus $R=A/I$, $A$ is a regular local ring of dimension $2$. But since $R$ is CM, $I$ must be of pure height one, and since $A$ is regular, $I$ is principal.

Note: in fact, the dimension one condition is not necessary, you can take a full length regular sequence and argue the same way (with just a little more work). So any CM singularity with multiplicity $2$ is a hypersurface.

EDIT: Once we know that the local ring at the singular point is a hypersurface, Jason Starr and Qing Liu's nice comments show that there must exist a smooth surface containing $C$, thus completely answer the question.

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  • $\begingroup$ The original question surely assumed this, but just for completeness I'll mention that you need the residue field to be infinite to get such an element x. $\endgroup$
    – Graham Leuschke
    Commented Jul 19, 2011 at 15:08
  • $\begingroup$ @Graham: thanks for catching that, we do need the residue field to be infinite. $\endgroup$
    – Hailong Dao
    Commented Jul 19, 2011 at 15:20
  • $\begingroup$ Thanks for the answer. Wouldn't this imply that at least one of the surfaces containing $C$ is smooth at the point of $C$ of multiplicity=2 ? $\endgroup$
    – mrw
    Commented Jul 19, 2011 at 16:54
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    $\begingroup$ Dear mrw -- To put Dao's argument another way, multiplicity 2 implies that the vector space $m_C/m_C^2$ is 2-dimensional. Since $m_X/m_X^2$ is 3-dimensional, there is a nonzero element $\overline{f}$ in the kernel of the restriction map $m_X/m_X^2 \to m_C/m_C^2$. But the kernel of this map is $I/(I+m_X^2)$, where $I$ is the kernel of the restriction map of coordinate algebras $k[X] \to k[C]$ (assume $X$ is affine). Therefore there is an element $f$ in $I$ whose image $\overline{f}$ in $m_X/m_X^2$ is nonzero, i.e., the zero set of $f$ contains $C$ and is nonsingular at your point. $\endgroup$
    – Jason Starr
    Commented Jul 19, 2011 at 17:41
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Being a local complete intersection is a "formal property", cf. Theorem 21.2, p. 171 of "Commutative Ring Theory" by H. Matsumura. So if you can find your smooth surface "infinitesimally" as you say, then your curve is a local complete intersection.

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