Limits of nodal surfaces and applications
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910929821630464 |
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| author | Ciliberto, Ciro Galati, Concettina |
| author_facet | Ciliberto, Ciro Galati, Concettina |
| contents | Let $\mathcal X\to\mathbb D$ be a flat family of projective complex 3-folds over a disc $\mathbb D$ with smooth total space $\mathcal X$ and smooth general fibre $\mathcal X_t,$ and whose special fiber $\mathcal X_0$ has double normal crossing singularities, in particular, $\mathcal X_0=A\cup B$, with $A$, $B$ smooth threefolds intersecting transversally along a smooth surface $R=A\cap B.$ In this paper we first study the limit singularities of a $δ$--nodal surface in the general fibre $S_t\subset\mathcal X_t$, when $S_t$ tends to the central fibre in such a way its $δ$ nodes tend to distinct points in $R$. The result is that the limit surface $S_0$ is in general the union $S_0=S_A\cup S_B$, with $S_A\subset A$, $S_B\subset B$ smooth surfaces, intersecting on $R$ along a $δ$-nodal curve $C=S_A\cap R=S_B\cap B$. Then we prove that, under suitable conditions, a surface $S_0=S_A\cup S_B$ as above indeed deforms to a $δ$--nodal surface in the general fibre of $\mathcal X\to\mathbb D$. As applications we prove that there are regular irreducible components of the Severi variety of degree $d$ surfaces with $δ$ nodes in $\mathbb P^3$, for every $δ\leq {d-1\choose 2}$ and of the Severi variety of complete intersection $δ$-nodal surfaces of type $(d,h)$, with $d\geq h-1$ in $\mathbb P^4$, for every $δ\leq {{d+3}\choose 3}-{{d-h+1}\choose 3}-1.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12365 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limits of nodal surfaces and applications Ciliberto, Ciro Galati, Concettina Algebraic Geometry 14B07, 14J17, 14C20 Let $\mathcal X\to\mathbb D$ be a flat family of projective complex 3-folds over a disc $\mathbb D$ with smooth total space $\mathcal X$ and smooth general fibre $\mathcal X_t,$ and whose special fiber $\mathcal X_0$ has double normal crossing singularities, in particular, $\mathcal X_0=A\cup B$, with $A$, $B$ smooth threefolds intersecting transversally along a smooth surface $R=A\cap B.$ In this paper we first study the limit singularities of a $δ$--nodal surface in the general fibre $S_t\subset\mathcal X_t$, when $S_t$ tends to the central fibre in such a way its $δ$ nodes tend to distinct points in $R$. The result is that the limit surface $S_0$ is in general the union $S_0=S_A\cup S_B$, with $S_A\subset A$, $S_B\subset B$ smooth surfaces, intersecting on $R$ along a $δ$-nodal curve $C=S_A\cap R=S_B\cap B$. Then we prove that, under suitable conditions, a surface $S_0=S_A\cup S_B$ as above indeed deforms to a $δ$--nodal surface in the general fibre of $\mathcal X\to\mathbb D$. As applications we prove that there are regular irreducible components of the Severi variety of degree $d$ surfaces with $δ$ nodes in $\mathbb P^3$, for every $δ\leq {d-1\choose 2}$ and of the Severi variety of complete intersection $δ$-nodal surfaces of type $(d,h)$, with $d\geq h-1$ in $\mathbb P^4$, for every $δ\leq {{d+3}\choose 3}-{{d-h+1}\choose 3}-1.$ |
| title | Limits of nodal surfaces and applications |
| topic | Algebraic Geometry 14B07, 14J17, 14C20 |
| url | https://arxiv.org/abs/2406.12365 |