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

Nasu, Hirokazu

Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1289-1316.

Nous étudions le schéma de Hilbert ${\mathrm{Hilb}}^{sc}V$ des courbes lisses connexes sur une variété de del Pezzo lisse $V$ de dimension 3. Nous montrons qu’aucune courbe $C$ dégénérée, c’est-à-dire, aucune courbe $C$ contenue dans une section hyperplane $S$ de $V$, se déforme en une courbe non-dégénérée, si les deux conditions suivantes sont satisfaites  : (i) $\chi (V,{ℐ}_C\left(S\right))\ge 1$ et (ii) pour chaque droite $\ell$ sur $S$ telle que $\ell \cap C=\varnothing$, le fibré normal $N_{\ell /V}$ de $\ell$ dans $V$ est trivial. Par conséquent, nous prouvons un analogue (pour ${\mathrm{Hilb}}^{sc}V$) d’une conjecture de J. O. Kleppe, qui concerne les composantes non-réduites du schéma de Hilbert ${\mathrm{Hilb}}^{sc}{\mathbb{P}}^3$ des courbes dans l’espace projectif ${\mathbb{P}}^3$ de dimension 3.

We study the Hilbert scheme ${\mathrm{Hilb}}^{sc}V$ 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) $\chi (V,{ℐ}_C\left(S\right))\ge 1$ and (ii) for every line $\ell$ on $S$ such that $\ell \cap C=\varnothing$, the normal bundle $N_{\ell /V}$ is trivial (i.e.  $N_{\ell /V}\simeq {{𝒪}_{{\mathbb{P}}^1}}^{\oplus 2}$). As a consequence, we prove an analogue (for ${\mathrm{Hilb}}^{sc}V$) of a conjecture of J. O. Kleppe, which is concerned with non-reduced components of the Hilbert scheme ${\mathrm{Hilb}}^{sc}{\mathbb{P}}^3$ of curves in the projective $3$-space ${\mathbb{P}}^3$.

Classification : 14C05, 14H10, 14D15
Keywords: Hilbert scheme, infinitesimal deformation, del Pezzo variety
Mot clés : schéma de Hilbert, déformations infinitésimales, variété de del Pezzo

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