On complete intersections containing a linear subspace
Consider the Fano scheme F k ( Y ) parameterizing k -dimensional linear subspaces contained in a complete intersection Y ⊂ P m of multi-degree d ̲ = ( d 1 , … , d s ) . It is known that, if t : = ∑ i = 1 s d i + k k - ( k + 1 ) ( m - k ) ⩽ 0 and ∏ i = 1 s d i > 2 , for Y a general complete inters...
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| Veröffentlicht in: | Geometriae dedicata 2020-02, Vol.204 (1), p.231-239 |
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| creator | Bastianelli, Francesco Ciliberto, Ciro Flamini, Flaminio Supino, Paola |
| description | Consider the Fano scheme
F
k
(
Y
)
parameterizing
k
-dimensional linear subspaces contained in a complete intersection
Y
⊂
P
m
of multi-degree
d
̲
=
(
d
1
,
…
,
d
s
)
. It is known that, if
t
:
=
∑
i
=
1
s
d
i
+
k
k
-
(
k
+
1
)
(
m
-
k
)
⩽
0
and
∏
i
=
1
s
d
i
>
2
, for
Y
a general complete intersection as above, then
F
k
(
Y
)
has dimension
-
t
. In this paper we consider the case
t
>
0
. Then the locus
W
d
̲
,
k
of all complete intersections as above containing a
k
-dimensional linear subspace is irreducible and turns out to have codimension
t
in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general
[
Y
]
∈
W
d
̲
,
k
the scheme
F
k
(
Y
)
is zero-dimensional of length one. This implies that
W
d
̲
,
k
is rational. |
| doi_str_mv | 10.1007/s10711-019-00452-2 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2343581299</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2343581299</sourcerecordid><originalsourceid>FETCH-LOGICAL-c270t-b4469458d45c1c650772b4e1f5f46dfd69ec2705883f43445f3f9e8f5496962b3</originalsourceid><addsrcrecordid>eNp9kEtLAzEUhYMoWKt_wNWA6-jNe7KU4gsK3eg6zKQ3ZUqbGZOZhf_e1BHcubpw-M658BFyy-CeAZiHzMAwRoFZCiAVp_yMLJgynFqm63OyKKmmyih1Sa5y3gOANYYviN7EyvfH4YAjVl0cMWX0Y9fHXOI4Nl3s4q5qqkMXsUlVnto8NB6vyUVoDhlvfu-SfDw_va9e6Xrz8rZ6XFPPDYy0lVJbqeqtVJ55raD8bCWyoILU27DVFk-gqmsRpJBSBREs1kFJq63mrViSu3l3SP3nhHl0-35Ksbx0XEihasatLRSfKZ_6nBMGN6Tu2KQvx8Cd_LjZjyt-3I-f0l4SMZdygeMO09_0P61vpIVmvA</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2343581299</pqid></control><display><type>article</type><title>On complete intersections containing a linear subspace</title><source>Springer Nature - Complete Springer Journals</source><creator>Bastianelli, Francesco ; Ciliberto, Ciro ; Flamini, Flaminio ; Supino, Paola</creator><creatorcontrib>Bastianelli, Francesco ; Ciliberto, Ciro ; Flamini, Flaminio ; Supino, Paola</creatorcontrib><description>Consider the Fano scheme
F
k
(
Y
)
parameterizing
k
-dimensional linear subspaces contained in a complete intersection
Y
⊂
P
m
of multi-degree
d
̲
=
(
d
1
,
…
,
d
s
)
. It is known that, if
t
:
=
∑
i
=
1
s
d
i
+
k
k
-
(
k
+
1
)
(
m
-
k
)
⩽
0
and
∏
i
=
1
s
d
i
>
2
, for
Y
a general complete intersection as above, then
F
k
(
Y
)
has dimension
-
t
. In this paper we consider the case
t
>
0
. Then the locus
W
d
̲
,
k
of all complete intersections as above containing a
k
-dimensional linear subspace is irreducible and turns out to have codimension
t
in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general
[
Y
]
∈
W
d
̲
,
k
the scheme
F
k
(
Y
)
is zero-dimensional of length one. This implies that
W
d
̲
,
k
is rational.</description><identifier>ISSN: 0046-5755</identifier><identifier>EISSN: 1572-9168</identifier><identifier>DOI: 10.1007/s10711-019-00452-2</identifier><language>eng</language><publisher>Dordrecht: Springer Netherlands</publisher><subject>Algebraic Geometry ; Convex and Discrete Geometry ; Differential Geometry ; Hyperbolic Geometry ; Intersections ; Mathematics ; Mathematics and Statistics ; Original Paper ; Projective Geometry ; Subspaces ; Topology</subject><ispartof>Geometriae dedicata, 2020-02, Vol.204 (1), p.231-239</ispartof><rights>Springer Nature B.V. 2019</rights><rights>2019© Springer Nature B.V. 2019</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-b4469458d45c1c650772b4e1f5f46dfd69ec2705883f43445f3f9e8f5496962b3</cites><orcidid>0000-0001-6111-8529</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10711-019-00452-2$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10711-019-00452-2$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,778,782,27907,27908,41471,42540,51302</link.rule.ids></links><search><creatorcontrib>Bastianelli, Francesco</creatorcontrib><creatorcontrib>Ciliberto, Ciro</creatorcontrib><creatorcontrib>Flamini, Flaminio</creatorcontrib><creatorcontrib>Supino, Paola</creatorcontrib><title>On complete intersections containing a linear subspace</title><title>Geometriae dedicata</title><addtitle>Geom Dedicata</addtitle><description>Consider the Fano scheme
F
k
(
Y
)
parameterizing
k
-dimensional linear subspaces contained in a complete intersection
Y
⊂
P
m
of multi-degree
d
̲
=
(
d
1
,
…
,
d
s
)
. It is known that, if
t
:
=
∑
i
=
1
s
d
i
+
k
k
-
(
k
+
1
)
(
m
-
k
)
⩽
0
and
∏
i
=
1
s
d
i
>
2
, for
Y
a general complete intersection as above, then
F
k
(
Y
)
has dimension
-
t
. In this paper we consider the case
t
>
0
. Then the locus
W
d
̲
,
k
of all complete intersections as above containing a
k
-dimensional linear subspace is irreducible and turns out to have codimension
t
in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general
[
Y
]
∈
W
d
̲
,
k
the scheme
F
k
(
Y
)
is zero-dimensional of length one. This implies that
W
d
̲
,
k
is rational.</description><subject>Algebraic Geometry</subject><subject>Convex and Discrete Geometry</subject><subject>Differential Geometry</subject><subject>Hyperbolic Geometry</subject><subject>Intersections</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Paper</subject><subject>Projective Geometry</subject><subject>Subspaces</subject><subject>Topology</subject><issn>0046-5755</issn><issn>1572-9168</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp9kEtLAzEUhYMoWKt_wNWA6-jNe7KU4gsK3eg6zKQ3ZUqbGZOZhf_e1BHcubpw-M658BFyy-CeAZiHzMAwRoFZCiAVp_yMLJgynFqm63OyKKmmyih1Sa5y3gOANYYviN7EyvfH4YAjVl0cMWX0Y9fHXOI4Nl3s4q5qqkMXsUlVnto8NB6vyUVoDhlvfu-SfDw_va9e6Xrz8rZ6XFPPDYy0lVJbqeqtVJ55raD8bCWyoILU27DVFk-gqmsRpJBSBREs1kFJq63mrViSu3l3SP3nhHl0-35Ksbx0XEihasatLRSfKZ_6nBMGN6Tu2KQvx8Cd_LjZjyt-3I-f0l4SMZdygeMO09_0P61vpIVmvA</recordid><startdate>20200201</startdate><enddate>20200201</enddate><creator>Bastianelli, Francesco</creator><creator>Ciliberto, Ciro</creator><creator>Flamini, Flaminio</creator><creator>Supino, Paola</creator><general>Springer Netherlands</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-6111-8529</orcidid></search><sort><creationdate>20200201</creationdate><title>On complete intersections containing a linear subspace</title><author>Bastianelli, Francesco ; Ciliberto, Ciro ; Flamini, Flaminio ; Supino, Paola</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-b4469458d45c1c650772b4e1f5f46dfd69ec2705883f43445f3f9e8f5496962b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Algebraic Geometry</topic><topic>Convex and Discrete Geometry</topic><topic>Differential Geometry</topic><topic>Hyperbolic Geometry</topic><topic>Intersections</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Paper</topic><topic>Projective Geometry</topic><topic>Subspaces</topic><topic>Topology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bastianelli, Francesco</creatorcontrib><creatorcontrib>Ciliberto, Ciro</creatorcontrib><creatorcontrib>Flamini, Flaminio</creatorcontrib><creatorcontrib>Supino, Paola</creatorcontrib><collection>CrossRef</collection><jtitle>Geometriae dedicata</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bastianelli, Francesco</au><au>Ciliberto, Ciro</au><au>Flamini, Flaminio</au><au>Supino, Paola</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On complete intersections containing a linear subspace</atitle><jtitle>Geometriae dedicata</jtitle><stitle>Geom Dedicata</stitle><date>2020-02-01</date><risdate>2020</risdate><volume>204</volume><issue>1</issue><spage>231</spage><epage>239</epage><pages>231-239</pages><issn>0046-5755</issn><eissn>1572-9168</eissn><abstract>Consider the Fano scheme
F
k
(
Y
)
parameterizing
k
-dimensional linear subspaces contained in a complete intersection
Y
⊂
P
m
of multi-degree
d
̲
=
(
d
1
,
…
,
d
s
)
. It is known that, if
t
:
=
∑
i
=
1
s
d
i
+
k
k
-
(
k
+
1
)
(
m
-
k
)
⩽
0
and
∏
i
=
1
s
d
i
>
2
, for
Y
a general complete intersection as above, then
F
k
(
Y
)
has dimension
-
t
. In this paper we consider the case
t
>
0
. Then the locus
W
d
̲
,
k
of all complete intersections as above containing a
k
-dimensional linear subspace is irreducible and turns out to have codimension
t
in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general
[
Y
]
∈
W
d
̲
,
k
the scheme
F
k
(
Y
)
is zero-dimensional of length one. This implies that
W
d
̲
,
k
is rational.</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s10711-019-00452-2</doi><tpages>9</tpages><orcidid>https://orcid.org/0000-0001-6111-8529</orcidid><oa>free_for_read</oa></addata></record> |
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| language | eng |
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| source | Springer Nature - Complete Springer Journals |
| subjects | Algebraic Geometry Convex and Discrete Geometry Differential Geometry Hyperbolic Geometry Intersections Mathematics Mathematics and Statistics Original Paper Projective Geometry Subspaces Topology |
| title | On complete intersections containing a linear subspace |
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