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If $C\subset\mathbb{P}^2$ is an integral curve of degree $d$, do its singularities deform independently as we vary $C$ over degree $d$ curves? If not, what about in the case $C$ is a nodal curve or $C$ is a general nodal curve?

Edit: In other words, is the composition $H^0(C,N_{C/\mathbb{P}^2})={\rm Hom}(N_{C/\mathbb{P}^2}^\vee,\mathscr{O}_C)\rightarrow {\rm Ext}^1(\Omega_C,\mathscr{O}_C)\rightarrow Ext^1(\Omega_C,\mathscr{O}_C)$ surjective?

(If I did the reduction correctly, which is not a given, I think this is equivalent to the vector space $V=k[X,Y,Z]_d$ of degree $d$ homogenous polynomials surjecting onto the sheaf associated to the graded module $k[X,Y,Z]/(F,\partial_XF,\partial_YF,\partial_ZF)$, where $C=V(F)$.)

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    $\begingroup$ For curves that are worse than nodal, this map is not surjective. For instance, consider the "hypercuspidal curve", $C=\text{Zero}(Y^d - X Z^{d-1})$. In this case, $\textit{Ext}^1 ( \Omega_C, \mathcal{O}_C )$ is a torsion sheaf supported at the unique singular point $[X,Y,Z] = [1,0,0]$ with length $(d-1)(d-2)$. On the other hand $k[X,Y,Z]_d$ has length $(d+2)(d+1)/2$. For $d\geq 9$, the map cannot be surjective. For the nodal case, you might check Harris's paper solving the Severi conjecture -- I remember something about this in that paper. $\endgroup$
    – Jason Starr
    Commented Aug 15, 2016 at 22:42
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    $\begingroup$ @JasonStarr: I think that the original work of Severi already answers the question about the possibility of independently smoothing the singularity of nodal curves. In fact, the proof of this result only involves checking the smoothness of some deformation space, and Severi's proof was in this aspect correct (as far I can remember). Harris solved the much harder problem of proving its irreducibility, for which Severi's proof was flawed. $\endgroup$
    – Francesco Polizzi
    Commented Aug 17, 2016 at 8:32

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A good survey on this problem is the paper by Greuel, Lossen and Shustin Equisingular families of projective curves, see http://arxiv.org/pdf/math/0612310.pdf.

In particular, at page 5 one can find the reference [Sev68] to the work of Severi showing that singular points of a nodal curve, irreducible or not, can be smoothed, or preserved, independently. This answers the last part of your question.

Regarding the possibility of independently smoothing singularities worse than nodes, as explained in J. Starr's comment the answer is in general no, even if this can be hard to check in explicit examples.

In fact, at p. 30 of the aforementioned paper the authors state the following more general question for cuspidal curves:

Are there $k$ and $d$ such that a curve of degree $d$ with $k$ cusps does exist, but with $k′ < k$ cusps does not ?

A candidate are a series of irreducible cuspidal curves constructed by Hirano (reference [Hir92]). They have degree $d= 2 \cdot 3^t$, with $t \geq 1$, and contain precisely $k=9(9t −1)/8$ cusps.

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  • $\begingroup$ Thanks, do you know where I can find the proof in the nodal case? I see the reference to Severi, but it's a 400-page German book, and I can't find where the argument is written. Or maybe there is a modern account that is easier to read? $\endgroup$
    – DCT
    Commented Aug 17, 2016 at 15:52

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