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Rendiconti Lincei - Matematica e Applicazioni

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Volume 17, Issue 2, 2006, pp. 95–123
DOI: 10.4171/RLM/457

Published online: 2006-06-30

Degenerations of scrolls to unions of planes

Alberto Calabri[1], Ciro Ciliberto[2], Flaminio Flamini[3] and Rick Miranda[4]

(1) Università di Ferrara, Italy
(2) Università di Roma, Italy
(3) Università di Roma Tor Vergata, Italy
(4) Colorado State University, Fort Collins, United States

In this paper we study degenerations of scrolls to union of planes, a problem already considered by G. Zappa in \cite{Zap1} and \cite{Zap2}. We prove, using techniques different from the ones of Zappa, a degeneration result to union of planes with the mildest possible singularities, for linearly normal scrolls of genus $g$ and of degree $d \geq 2g+4$ in $\Pp^{d-2g+1}$. We also study properties of components of the Hilbert scheme parametrizing scrolls. Finally we review Zappa's original approach.

Keywords: Ruled surfaces, embedded degenerations, Hilbert schemes of scrolls, moduli

Calabri Alberto, Ciliberto Ciro, Flamini Flaminio, Miranda Rick: Degenerations of scrolls to unions of planes. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 17 (2006), 95-123. doi: 10.4171/RLM/457