Limits of nodal surfaces and applications
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 singular...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000374/type/journal_article |
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| Summary: | 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
$\delta $
-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
$\delta $
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
$\delta $
-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
$\delta $
-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
$\delta $
nodes in
$\mathbb {P}^3$
, for every
$\delta \leqslant {d-1\choose 2}$
and of the Severi variety of complete intersection
$\delta $
-nodal surfaces of type
$(d,h)$
, with
$d\geqslant h-1$
in
$\mathbb {P}^4$
, for every
$\delta \leqslant {{d+3}\choose 3}-{{d-h+1}\choose 3}-1.$
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| ISSN: | 2050-5094 |