Let $P_1,P$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Denote by $H_{P_1,P}$ the flag Hilbert scheme parametrizing pairs $(C_1 \subset C)$ where $C_1, C$ are of Hilbert polynomials $P_1$ and $P$, respectively. Let $L$ be an irreducible component of $H_{P_1,P}$. Assume that a general element of $L$ is a pair $(C_1 \subset C)$ where $C_1, C$ are local complete intersection curves in $\mathbb{P}^3$. Denote by $P_2$ the Hilbert polynomial of the residual curve $C_2$ (i.e., $C_1 \cup C_2 = C$). Assume that $C_2$ is a local complete intersection curve. The question is does there exist an irreducible component $L'$ in $H_{P_2,P}$ such that for a general pair $(D_2 \subset D)$ the residual curve $D_1$ (i.e., $D_1 \cup D_2 =D$) satisfies: $(D_1 \subset D)$ is an element in $L$? The motivation for this question is that a similar result holds true in the case $C$ is a complete intersection curve.
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$\begingroup$ Of course the basic difficulty, as your title suggests, is that, although $C_2$ is Cohen-Macaulay, it need not be Gorenstein. For instance, if $C$ is (formally locally) a union of four concurrent lines and $C_1$ is one of those four lines, then the residual $C_2$ is not Gorenstein. Given a flat deformation of $(C,C_2)$ to $(D,D_2)$, if $D_2$ is Gorenstein, then the residual $D_1$ is also flat. However, I believe this can fail otherwise. This should be (intimately) connected to failure of flatness for the relative dualizing sheaf of a flat family of Cohen-Macaulay (non-Gorenstein) curves. $\endgroup$– Jason StarrJul 25, 2014 at 13:45
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$\begingroup$ @Starr: Thank you for the answer. I may be wrong, but I thought local complete intersection curves $X$ in $\mathbb{P}^3$ are Gorenstein, since for any closed point $x \in X$, $\mathcal{O}_{X,x}$ is the quotient of a regular ring by a regular sequence, hence a Gorenstein ring (several texts mention that l.c.i. is Gorenstein without giving proofs, including E. Sernesi's books on Deformation of Algebraic schemes). $\endgroup$– KaliJul 25, 2014 at 14:04
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$\begingroup$ Yes, local complete intersections are Gorenstein. However, just because $C$ and $C_1$ are local complete intersections (and thus Gorenstein), that does not imply that the residual $C_2$ is Gorenstein. $\endgroup$– Jason StarrJul 25, 2014 at 16:06
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$\begingroup$ @Starr: Thank you. Could you please suggest some reference which studies such questions i.e., under what additional conditions the residual curve is Gorenstein? $\endgroup$– KaliJul 25, 2014 at 16:16
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$\begingroup$ One reference is Theorem 21.23, p. 546, of Eisenbud's "Commutative Algebra with a View Toward Algebraic Geometry". $\endgroup$– Jason StarrJul 25, 2014 at 16:30
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