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Obstructions to deforming curves on a $3$ -fold, II: Deformations of degenerate curves on a del Pezzo $3$ -fold

Hirokazu Nasu (2010)

Annales de l’institut Fourier

We study the Hilbert scheme ${Hilb}^{s c} V$ of smooth connected curves on a smooth del Pezzo $3$ -fold $V$ . We prove that any degenerate curve $C$ , i.e. any curve $C$ contained in a smooth hyperplane section $S$ of $V$ , does not deform to a non-degenerate curve if the following two conditions are satisfied: (i) $χ (V, ℐ_{C} (S)) \geq 1$ and (ii) for every line $ℓ$ on $S$ such that $ℓ \cap C = \emptyset$ , the normal bundle $N_{ℓ / V}$ is trivial (i.e. $N_{ℓ / V} ≃ {𝒪_{ℙ^{1}}}^{\oplus 2}$ ). As a consequence, we prove an analogue (for ${Hilb}^{s c} V$ ) of a conjecture of J. O. Kleppe, which is concerned with non-reduced components...

On a Theorem of Castelnuovo, and the Equations Defining Space Curves.

L. Gruson, C. Peskine, R. Lazarsfeld (1983)

Inventiones mathematicae

On algebraic points on curves

Paul Vojta (1991)

Compositio Mathematica

On Castelnuovo's equivalence defect.

Ernst Kani (1984)

Journal für die reine und angewandte Mathematik

On curves with a theta-characteristic whose space of sections has dimension 4.

Elisabetta Colombo (1994)

Mathematische Zeitschrift

On Dimension Theorems of the Varieties of Special Divisors on a Curve.

G. Martens (1984)

Mathematische Annalen

On Linearly Normal Space Curves.

A. Dolcetti, G. Pareschi (1988)

Mathematische Zeitschrift

On linearly normal strange curves

Edoardo Ballico (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Here we prove a numerical bound implying that, except for smooth plane conics in characteristic 2, no complete linear system maps birationally a smooth curve into a projective space with a strange curve as image.