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...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
|
| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000374/type/journal_article |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850172390173048832 |
|---|---|
| author | Ciro Ciliberto Concettina Galati |
| author_facet | Ciro Ciliberto Concettina Galati |
| author_sort | Ciro Ciliberto |
| collection | DOAJ |
| description | 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
$\delta \leqslant {{d+3}\choose 3}-{{d-h+1}\choose 3}-1.$
|
| format | Article |
| id | doaj-art-3d81b4e983f34130887eaca1ae7e977e |
| institution | OA Journals |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-3d81b4e983f34130887eaca1ae7e977e2025-08-20T02:20:06ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.37Limits of nodal surfaces and applicationsCiro Ciliberto0https://orcid.org/0000-0002-3060-1056Concettina Galati1https://orcid.org/0000-0003-0016-8201Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, Roma, 00133, Italy; E-mail:Dipartimento di Matematica e Informatica, Università della Calabria, via P. Bucci, cubo 31B, Arcavacata di Rende (CS), 87036, ItalyLet $\mathcal {X}\to \mathbb {D}$ 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 $\delta \leqslant {{d+3}\choose 3}-{{d-h+1}\choose 3}-1.$ https://www.cambridge.org/core/product/identifier/S2050509425000374/type/journal_article14B0714J1714C20 |
| spellingShingle | Ciro Ciliberto Concettina Galati Limits of nodal surfaces and applications Forum of Mathematics, Sigma 14B07 14J17 14C20 |
| title | Limits of nodal surfaces and applications |
| title_full | Limits of nodal surfaces and applications |
| title_fullStr | Limits of nodal surfaces and applications |
| title_full_unstemmed | Limits of nodal surfaces and applications |
| title_short | Limits of nodal surfaces and applications |
| title_sort | limits of nodal surfaces and applications |
| topic | 14B07 14J17 14C20 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425000374/type/journal_article |
| work_keys_str_mv | AT cirociliberto limitsofnodalsurfacesandapplications AT concettinagalati limitsofnodalsurfacesandapplications |