Extension groups of tautological bundles on symmetric products of curves
We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if E ≠ O X is simple, then the natural map Ext 1 ( E , E ) → Ext 1 ( E [ n ] , E [ n ] ) is injective for every n . Along with previous results, this i...
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| Veröffentlicht in: | Beiträge zur Algebra und Geometrie 2023-06, Vol.64 (2), p.493-530 |
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| creator | Krug, Andreas |
| description | We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if
E
≠
O
X
is simple, then the natural map
Ext
1
(
E
,
E
)
→
Ext
1
(
E
[
n
]
,
E
[
n
]
)
is injective for every
n
. Along with previous results, this implies that
E
↦
E
[
n
]
defines an embedding of the moduli space of stable bundles of slope
μ
∉
[
-
1
,
n
-
1
]
on the curve
X
into the moduli space of stable bundles on the symmetric product
X
(
n
)
. The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of
X
with the loci in the moduli space of stable bundles on
X
(
n
)
where the dimension of the tangent space jumps. We also prove that
E
[
n
]
is simple if
E
is simple. |
| doi_str_mv | 10.1007/s13366-022-00644-0 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2807041828</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2807041828</sourcerecordid><originalsourceid>FETCH-LOGICAL-c363t-fb5b66a47362502681cead12814a07349cc33a77662f6b8f928d9bae19f2ef5d3</originalsourceid><addsrcrecordid>eNp9kE1LAzEQhoMoWKt_wNOC5-gk2SbZo5RqhYIXPYdsNilbtpuaD7H_3tgVvHkaGJ73neFB6JbAPQEQD5EwxjkGSjEAr2sMZ2hGSUMwMMnO0QwIk7iWlFyiqxh3UCghxAytV1_JjrH3Y7UNPh9i5V2VdE5-8Nve6KFq89gNtuzHKh73e5tCb6pD8F026USbHD5tvEYXTg_R3vzOOXp_Wr0t13jz-vyyfNxgwzhL2LWLlnNdC8bpAiiXxFjdESpJrUGwujGGMS0E59TxVrqGyq5ptSWNo9YtOjZHd1NveeEj25jUzucwlpOKShBQE0lloehEmeBjDNapQ-j3OhwVAfVjTE3GVDGmTsYUlBCbQrHA49aGv-p_Ut-Zhm6I</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2807041828</pqid></control><display><type>article</type><title>Extension groups of tautological bundles on symmetric products of curves</title><source>Springer Nature - Complete Springer Journals</source><creator>Krug, Andreas</creator><creatorcontrib>Krug, Andreas</creatorcontrib><description>We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if
E
≠
O
X
is simple, then the natural map
Ext
1
(
E
,
E
)
→
Ext
1
(
E
[
n
]
,
E
[
n
]
)
is injective for every
n
. Along with previous results, this implies that
E
↦
E
[
n
]
defines an embedding of the moduli space of stable bundles of slope
μ
∉
[
-
1
,
n
-
1
]
on the curve
X
into the moduli space of stable bundles on the symmetric product
X
(
n
)
. The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of
X
with the loci in the moduli space of stable bundles on
X
(
n
)
where the dimension of the tangent space jumps. We also prove that
E
[
n
]
is simple if
E
is simple.</description><identifier>ISSN: 0138-4821</identifier><identifier>EISSN: 2191-0383</identifier><identifier>DOI: 10.1007/s13366-022-00644-0</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Algebra ; Algebraic Geometry ; Convex and Discrete Geometry ; Embedding ; Geometry ; Loci ; Mathematics ; Mathematics and Statistics ; Original Paper</subject><ispartof>Beiträge zur Algebra und Geometrie, 2023-06, Vol.64 (2), p.493-530</ispartof><rights>The Author(s) 2022</rights><rights>The Author(s) 2022. This work is published under http://creativecommons.org/licenses/by/4.0/. (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-fb5b66a47362502681cead12814a07349cc33a77662f6b8f928d9bae19f2ef5d3</citedby><cites>FETCH-LOGICAL-c363t-fb5b66a47362502681cead12814a07349cc33a77662f6b8f928d9bae19f2ef5d3</cites><orcidid>0000-0003-1956-7368</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/s13366-022-00644-0$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s13366-022-00644-0$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Krug, Andreas</creatorcontrib><title>Extension groups of tautological bundles on symmetric products of curves</title><title>Beiträge zur Algebra und Geometrie</title><addtitle>Beitr Algebra Geom</addtitle><description>We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if
E
≠
O
X
is simple, then the natural map
Ext
1
(
E
,
E
)
→
Ext
1
(
E
[
n
]
,
E
[
n
]
)
is injective for every
n
. Along with previous results, this implies that
E
↦
E
[
n
]
defines an embedding of the moduli space of stable bundles of slope
μ
∉
[
-
1
,
n
-
1
]
on the curve
X
into the moduli space of stable bundles on the symmetric product
X
(
n
)
. The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of
X
with the loci in the moduli space of stable bundles on
X
(
n
)
where the dimension of the tangent space jumps. We also prove that
E
[
n
]
is simple if
E
is simple.</description><subject>Algebra</subject><subject>Algebraic Geometry</subject><subject>Convex and Discrete Geometry</subject><subject>Embedding</subject><subject>Geometry</subject><subject>Loci</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Paper</subject><issn>0138-4821</issn><issn>2191-0383</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9kE1LAzEQhoMoWKt_wNOC5-gk2SbZo5RqhYIXPYdsNilbtpuaD7H_3tgVvHkaGJ73neFB6JbAPQEQD5EwxjkGSjEAr2sMZ2hGSUMwMMnO0QwIk7iWlFyiqxh3UCghxAytV1_JjrH3Y7UNPh9i5V2VdE5-8Nve6KFq89gNtuzHKh73e5tCb6pD8F026USbHD5tvEYXTg_R3vzOOXp_Wr0t13jz-vyyfNxgwzhL2LWLlnNdC8bpAiiXxFjdESpJrUGwujGGMS0E59TxVrqGyq5ptSWNo9YtOjZHd1NveeEj25jUzucwlpOKShBQE0lloehEmeBjDNapQ-j3OhwVAfVjTE3GVDGmTsYUlBCbQrHA49aGv-p_Ut-Zhm6I</recordid><startdate>20230601</startdate><enddate>20230601</enddate><creator>Krug, Andreas</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0003-1956-7368</orcidid></search><sort><creationdate>20230601</creationdate><title>Extension groups of tautological bundles on symmetric products of curves</title><author>Krug, Andreas</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-fb5b66a47362502681cead12814a07349cc33a77662f6b8f928d9bae19f2ef5d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Algebra</topic><topic>Algebraic Geometry</topic><topic>Convex and Discrete Geometry</topic><topic>Embedding</topic><topic>Geometry</topic><topic>Loci</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Paper</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Krug, Andreas</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><jtitle>Beiträge zur Algebra und Geometrie</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Krug, Andreas</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Extension groups of tautological bundles on symmetric products of curves</atitle><jtitle>Beiträge zur Algebra und Geometrie</jtitle><stitle>Beitr Algebra Geom</stitle><date>2023-06-01</date><risdate>2023</risdate><volume>64</volume><issue>2</issue><spage>493</spage><epage>530</epage><pages>493-530</pages><issn>0138-4821</issn><eissn>2191-0383</eissn><abstract>We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if
E
≠
O
X
is simple, then the natural map
Ext
1
(
E
,
E
)
→
Ext
1
(
E
[
n
]
,
E
[
n
]
)
is injective for every
n
. Along with previous results, this implies that
E
↦
E
[
n
]
defines an embedding of the moduli space of stable bundles of slope
μ
∉
[
-
1
,
n
-
1
]
on the curve
X
into the moduli space of stable bundles on the symmetric product
X
(
n
)
. The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of
X
with the loci in the moduli space of stable bundles on
X
(
n
)
where the dimension of the tangent space jumps. We also prove that
E
[
n
]
is simple if
E
is simple.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s13366-022-00644-0</doi><tpages>38</tpages><orcidid>https://orcid.org/0000-0003-1956-7368</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0138-4821 |
| ispartof | Beiträge zur Algebra und Geometrie, 2023-06, Vol.64 (2), p.493-530 |
| issn | 0138-4821 2191-0383 |
| language | eng |
| recordid | cdi_proquest_journals_2807041828 |
| source | Springer Nature - Complete Springer Journals |
| subjects | Algebra Algebraic Geometry Convex and Discrete Geometry Embedding Geometry Loci Mathematics Mathematics and Statistics Original Paper |
| title | Extension groups of tautological bundles on symmetric products of curves |
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