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Let $X$ to be an irreducible cubic surface in $\mathbb{P}^3_{\mathbb{C}}$. Is it true that if $\dim(\operatorname{Sing}(X))\geq 1$, then a line is contained in $\operatorname{Sing}(X)$? I.e, does a cubic irreducible surface with a singular curve has a double line?

This may be a silly question, and already proved, but I'm searching the literature for an answer to it. It is not clear to me if it is true. Furthermore, I am not sure how would attempt to answer this question myself.

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  • $\begingroup$ You searching the literature for an answer to what? $\endgroup$
    – Sasha
    Commented May 27 at 20:08
  • $\begingroup$ @Sasha if this statement holds true. It is not clear to me $\endgroup$
    – ben huni
    Commented May 27 at 20:10
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    $\begingroup$ See [Miles Reid. Nonnormal del Pezzo surfaces. Publ. Res. Inst. Math. Sci., 30(5):695–727, 1994]. $\endgroup$
    – Sasha
    Commented May 27 at 20:20
  • $\begingroup$ @Sasha sorry, I have made some edits to my initial question. I hope this is more clear $\endgroup$
    – ben huni
    Commented May 27 at 20:52
  • $\begingroup$ I see the difficulty in stating this problem now. Thanks @Sasha $\endgroup$
    – ben huni
    Commented May 27 at 21:24

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