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Hodge Theory of Hyperplane Sections ...

Brogan, Daniel,

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Hodge Theory of Hyperplane Sections and Secant Varieties of Curves /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Hodge Theory of Hyperplane Sections and Secant Varieties of Curves // Daniel Brogan.
Author:	Brogan, Daniel,
Description:	1 electronic resource (76 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
Contained By:	Dissertations Abstracts International86-12B.
Subject:	Mathematics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=32041658
ISBN:	9798286438969

Hodge Theory of Hyperplane Sections and Secant Varieties of Curves /
Brogan, Daniel,

Hodge Theory of Hyperplane Sections and Secant Varieties of Curves /Daniel Brogan. - 1 electronic resource (76 pages)

Source: Dissertations Abstracts International, Volume: 86-12, Section: B.

This dissertation contains results of two research projects. In the first, we review the proof of Nori's connectivity theorem, a result usually applied in the study of algebraic cycles, and use a key result contained therein to study the Hodge module induced by the complete variation of Hodge structure associated to large degree hyperplane sections in a fixed smooth projective variety. This allows us to compute parts of the de Rahm complex in terms of the cohomology of the ambient variety. The second project is concerned with secant varieties of smooth projective curves. These are singular projective varieties, and we aim to understand their singularities by way of Hodge theory. We study the trivial mixed Hodge modules and intersection complexes of these varieties. For an arbitrary curve, we compute their intersection cohomology. In the case of a rational normal curve, we prove that the secant varieties are rational homology manifolds whose cohomology is of Hodge-Tate type. The largest secant variety of a rational normal curve is defined by the vanishing of the determinant of a generic Hankel matrix. Using the special geometry associated to this matrix, we compute the nearby and vanishing cycle sheaves with respect to this defining function.

English

ISBN: 9798286438969Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Hodge theory

Hodge Theory of Hyperplane Sections and Secant Varieties of Curves /
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245 1 0 $a Hodge Theory of Hyperplane Sections and Secant Varieties of Curves / $c Daniel Brogan.
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300 $a 1 electronic resource (76 pages)
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500 $a Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
500 $a Advisors: Schnell, Christian Committee members: de Cataldo, Mark; Laza, Radu; Mustata, Mircea.
502 $b Ph.D. $c State University of New York at Stony Brook $d 2025.
520 $a This dissertation contains results of two research projects. In the first, we review the proof of Nori's connectivity theorem, a result usually applied in the study of algebraic cycles, and use a key result contained therein to study the Hodge module induced by the complete variation of Hodge structure associated to large degree hyperplane sections in a fixed smooth projective variety. This allows us to compute parts of the de Rahm complex in terms of the cohomology of the ambient variety. The second project is concerned with secant varieties of smooth projective curves. These are singular projective varieties, and we aim to understand their singularities by way of Hodge theory. We study the trivial mixed Hodge modules and intersection complexes of these varieties. For an arbitrary curve, we compute their intersection cohomology. In the case of a rational normal curve, we prove that the secant varieties are rational homology manifolds whose cohomology is of Hodge-Tate type. The largest secant variety of a rational normal curve is defined by the vanishing of the determinant of a generic Hankel matrix. Using the special geometry associated to this matrix, we compute the nearby and vanishing cycle sheaves with respect to this defining function.
546 $a English
590 $a School code: 0771
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
653 $a Hodge theory
653 $a Hyperplane sections
653 $a Secant varieties
653 $a Smooth projective curves
653 $a Rahm complex
690 $a 0405
690 $a 0642
690 $a 0364
710 2 $a State University of New York at Stony Brook. $b Mathematics. $e degree granting institution. $3 3823091
720 1 $a Schnell, Christian $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 86-12B.
790 $a 0771
791 $a Ph.D.
792 $a 2025
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=32041658