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Complete Homogeneous Varieties via Representation Theory.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Complete Homogeneous Varieties via Representation Theory./
Author:	Cavazzani, Francesco.
Description:	1 online resource (97 pages)
Notes:	Source: Dissertation Abstracts International, Volume: 78-12(E), Section: B.
Contained By:	Dissertation Abstracts International78-12B(E).
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9780355030136

Complete Homogeneous Varieties via Representation Theory.
Cavazzani, Francesco.

Complete Homogeneous Varieties via Representation Theory. - 1 online resource (97 pages)

Source: Dissertation Abstracts International, Volume: 78-12(E), Section: B.

Thesis (Ph.D.)--Harvard University, 2016.

Includes bibliographical references

Given an algebraic variety X ⊂ P N with stabilizer H, the quotient PGLN+1/H can be interpreted a parameter space for all PGLN+1-translates of X. We define X to be a homogeneous variety if H acts on it transitively, and satisfies a few other properties, such as H being semisimple. Some examples of homogeneous varieties are quadric hypersurfaces, rational normal curves, and Veronese and Segre embeddings. In this case, we construct new compactifications of the parameter spaces PGLN+1/H, obtained compactifying PGLN+1 to the classically known space of complete collineations, and taking the G.I.T. quotient by H, and we will call the result space of complete homogeneous varieties; this extends the same construction for quadric hypersurfaces by Kannan in 1999. We establish a few properties of these spaces: in particular, we find a formula for the volume of divisors that depends only on the dimension of H-invariants in irreducible representations of SLN+1. We then develop some tools in invariant theory, combinatorics and spline approximation to calculate such invariants, and carry out the entire calculations for the case of SL2-invariants in irreducible representations of SL4, that gives us explicit values for the volume function in the case of X being a twisted cubic. Afterwards, we focus our attention on the case of twisted cubics, giving a more explicit description of these compactifications, including the relation with the previously known moduli spaces. In the end, we make some conjectures about how the volume function might be used in solving some enumerative problems.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2018

Mode of access: World Wide Web

ISBN: 9780355030136Subjects--Topical Terms:

527692
Mathematics.
Index Terms--Genre/Form:

554714
Electronic books.

Complete Homogeneous Varieties via Representation Theory.
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245 1 0 $a Complete Homogeneous Varieties via Representation Theory.
264 0 $c 2016
300 $a 1 online resource (97 pages)
336 $a text $b txt $2 rdacontent
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338 $a online resource $b cr $2 rdacarrier
500 $a Source: Dissertation Abstracts International, Volume: 78-12(E), Section: B.
500 $a Advisers: Joe Harris; Barry Mazur; Maksym Fedorchuk.
502 $a Thesis (Ph.D.)--Harvard University, 2016.
504 $a Includes bibliographical references
520 $a Given an algebraic variety X ⊂ P N with stabilizer H, the quotient PGLN+1/H can be interpreted a parameter space for all PGLN+1-translates of X. We define X to be a homogeneous variety if H acts on it transitively, and satisfies a few other properties, such as H being semisimple. Some examples of homogeneous varieties are quadric hypersurfaces, rational normal curves, and Veronese and Segre embeddings. In this case, we construct new compactifications of the parameter spaces PGLN+1/H, obtained compactifying PGLN+1 to the classically known space of complete collineations, and taking the G.I.T. quotient by H, and we will call the result space of complete homogeneous varieties; this extends the same construction for quadric hypersurfaces by Kannan in 1999. We establish a few properties of these spaces: in particular, we find a formula for the volume of divisors that depends only on the dimension of H-invariants in irreducible representations of SLN+1. We then develop some tools in invariant theory, combinatorics and spline approximation to calculate such invariants, and carry out the entire calculations for the case of SL2-invariants in irreducible representations of SL4, that gives us explicit values for the volume function in the case of X being a twisted cubic. Afterwards, we focus our attention on the case of twisted cubics, giving a more explicit description of these compactifications, including the relation with the previously known moduli spaces. In the end, we make some conjectures about how the volume function might be used in solving some enumerative problems.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2018
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 527692
655 7 $a Electronic books. $2 local $3 554714
690 $a 0405
710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a Harvard University. $b Mathematics. $3 1193029
773 0 $t Dissertation Abstracts International $g 78-12B(E).
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10632935 $z click for full text (PQDT)