Let $V$ be the Veronese surface, obtained as the image of $\mathbb{P}^2$ in $\mathbb{P}^5$ by the complete linear system of conics. I understand that $V$ can degenerate to the union of a cubic scroll $S(1,2)$ and a 2-plane $P\subset \mathbb{P}^5$, and there are cases where $S(1,2)\cap P$ is a line. Can $V$ degenerate to a case where $S(1,2)$ and $P$ intersect in just one reduced point?
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1$\begingroup$ A hyperplane section of a Veronese surface is a Veronese 2-uple image of a plane conic, i.e., a quartic rational curve. By the principle of connectedness, every specialization is a connected curve. Yet a general hyperplane section of your scheme is disconnected. $\endgroup$– Jason StarrCommented Aug 26 at 23:43
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The Hilbert polynomials of a Veronese surface, a cubic scroll, and a plane, are $$ P_1(t) = 2t^2 + 3t + 1, \quad P_2(t) = \frac32 t^2 + \frac52 t + 1, \quad P_3(t) = \frac12 t^2 + \frac32 t + 1, $$ respectively. Note that $$ P_2 + P_3 - P_1 = t + 1, $$ so if a Veronese surface (flatly) degenerates into the union of a cubic scroll and a plane intersecting along a scheme $Z$, then the Hilbert polynomial of $Z$ is $t + 1$. In particular, $Z$ must be a line.
