Obstructions to deforming space curves lying on a del Pezzo surface
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| Format: | Preprint |
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2025
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| _version_ | 1866912205457326080 |
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| author | Nasu, Hirokazu |
| author_facet | Nasu, Hirokazu |
| contents | We study the deformations of space curves $C \subset \mathbb P^4$, assuming that they are contained in a smooth complete intersection $S_{2,2} \subset \mathbb P^4$, i.e., a smooth del Pezzo surface of degree $4$. We give sufficient conditions for $C$ to be (un)obstructed in terms of the degree $d$ and the genus $g$ of $C$. We prove that if $d>8$, $g\ge 2d-12$, and $h^1(C,\mathcal I_C(2))=1$, then $C$ is obstructed and stably degenerate, i.e., $C$ has some first order infinitesimal deformations in $\mathbb P^4$ not contained in any deformations of $S_{2,2}$ in $\mathbb P^4$, but they do not lift to any global deformations. (As a result, every global deformation of $C$ in $\mathbb P^4$ is contained in a deformation of $S_{2,2}$ in $\mathbb P^4$.) As an application, we construct infinitely many examples of irreducible components of the Hilbert scheme $\operatorname{Hilb}^{sc} \mathbb P^4$ of smooth connected curves in $\mathbb P^4$, along which $\operatorname{Hilb}^{sc} \mathbb P^4$ is generically non-reduced. In the case $d=14$ and $g=16$, we obtain a non-reduced component of $\operatorname{Hilb}^{sc} \mathbb P^4$ of dimension $55$ with $\dim T_{\operatorname{Hilb}^{sc} \mathbb P^4}=57$, analogous to Mumford's example of a non-reduced component of $\operatorname{Hilb}^{sc} \mathbb P^3$, whose general member is contained in a smooth cubic surface $S_3 \subset \mathbb P^3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_15788 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Obstructions to deforming space curves lying on a del Pezzo surface Nasu, Hirokazu Algebraic Geometry 14C05, 14D15, 14H50 We study the deformations of space curves $C \subset \mathbb P^4$, assuming that they are contained in a smooth complete intersection $S_{2,2} \subset \mathbb P^4$, i.e., a smooth del Pezzo surface of degree $4$. We give sufficient conditions for $C$ to be (un)obstructed in terms of the degree $d$ and the genus $g$ of $C$. We prove that if $d>8$, $g\ge 2d-12$, and $h^1(C,\mathcal I_C(2))=1$, then $C$ is obstructed and stably degenerate, i.e., $C$ has some first order infinitesimal deformations in $\mathbb P^4$ not contained in any deformations of $S_{2,2}$ in $\mathbb P^4$, but they do not lift to any global deformations. (As a result, every global deformation of $C$ in $\mathbb P^4$ is contained in a deformation of $S_{2,2}$ in $\mathbb P^4$.) As an application, we construct infinitely many examples of irreducible components of the Hilbert scheme $\operatorname{Hilb}^{sc} \mathbb P^4$ of smooth connected curves in $\mathbb P^4$, along which $\operatorname{Hilb}^{sc} \mathbb P^4$ is generically non-reduced. In the case $d=14$ and $g=16$, we obtain a non-reduced component of $\operatorname{Hilb}^{sc} \mathbb P^4$ of dimension $55$ with $\dim T_{\operatorname{Hilb}^{sc} \mathbb P^4}=57$, analogous to Mumford's example of a non-reduced component of $\operatorname{Hilb}^{sc} \mathbb P^3$, whose general member is contained in a smooth cubic surface $S_3 \subset \mathbb P^3$. |
| title | Obstructions to deforming space curves lying on a del Pezzo surface |
| topic | Algebraic Geometry 14C05, 14D15, 14H50 |
| url | https://arxiv.org/abs/2501.15788 |