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Let $C \subset \mathbb{P}^n$ be an one dimensional scheme. Suppose that $C$ decomposes as the union of a Cohen Macaulay reduced curve $\tilde{C}$ (in particular $\tilde{C}$ does not have embedded points) and a finite number of embedded points $l_p$ such that $C= \tilde{C} \cup l_p$. Are the points $l_P$ necessarily located at singular points of $\tilde{C}$ ? I read a similar statement somewhere, but I cannot recall it or find the reference...

Thanks

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    $\begingroup$ Your statement is not true, take $C\subset \mathbb{P}^2$ defined by $x^2=xy=0$. This gives the line $x=0$ with an embedded point at the origin. $\endgroup$
    – abx
    Commented Aug 10, 2014 at 19:03
  • $\begingroup$ Thank you for the example. If you know of a weaker statement or a restriction on the support of $l_p$... I will appreciated it. $\endgroup$
    – NotNow
    Commented Aug 10, 2014 at 19:07
  • $\begingroup$ Can you say some more information about how $C$ is given to you? Is it presented to you via a certain number of relations? Is it the image of something? Of course as Cantlog was saying, the embedded points are singular points of $C$ but I'm not sure how to say much more than that with the information given. $\endgroup$
    – Karl Schwede
    Commented Aug 10, 2014 at 20:12
  • $\begingroup$ $l_p$ is necessarily a singular point of $C$.... $\endgroup$
    – Will Sawin
    Commented Aug 10, 2014 at 22:05
  • $\begingroup$ I am thinking in constructing curves $C$ with a given Hilbert polynomial by adding embedded points to another curve $\tilde{C}$. For example: Given a smooth plane cubic $\tilde{C}$ can I add an appropriate embedded point $l_P$ for obtaining a curve $C$ parametrized by the Hilbert scheme associated to $3t+1-d$ for any $d >0$? Can I add the point $l_p$ anywhere on $\tilde{C}$? $\endgroup$
    – NotNow
    Commented Aug 10, 2014 at 23:23

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