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Intersection Theory of the Moduli Sp...

Kong, Bochao.

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Intersection Theory of the Moduli Space of Elliptic K3 Surfaces.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Intersection Theory of the Moduli Space of Elliptic K3 Surfaces./
Author:	Kong, Bochao.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2024,
Description:	135 p.
Notes:	Source: Dissertations Abstracts International, Volume: 86-01, Section: B.
Contained By:	Dissertations Abstracts International86-01B.
Subject:	Mathematics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31300202
ISBN:	9798383219294

Intersection Theory of the Moduli Space of Elliptic K3 Surfaces.
Kong, Bochao.

Intersection Theory of the Moduli Space of Elliptic K3 Surfaces. - Ann Arbor : ProQuest Dissertations & Theses, 2024 - 135 p.

Source: Dissertations Abstracts International, Volume: 86-01, Section: B.

Thesis (Ph.D.)--University of California, San Diego, 2024.

Moduli spaces of K3 surfaces are fundamental objects in algebraic geometry. Elliptic K3 surfaces are K3 surfaces with elliptic fibration structure, and they are of particular interest due to their rich geometry. The moduli space of elliptic K3 surfaces can be studied using the theory of Weierstrass models. In this dissertation, we study the topology and intersection theory of the moduli space of elliptic K3 surfaces.We compute the Poincare polynomial of the moduli space of elliptic K3 surfaces. The main idea is constructing a compactification using the Weierstrass models, this compactification is a GIT quotient. We adapt Kirwan's blowup machinery to weighted projective space to compute the Poincare polynomial. We find the cohomology is mostly concentrated in the even degrees, but there is one odd degree class in degree 33.We also study the Chow ring of the moduli space of elliptic surfaces of degree N ≥ 2. We conclude that the Chow ring of the moduli space of elliptic surfaces is always generated by two classes. Furthermore, explicit relations between these classes are given, the Poincare polynomial for the Chow ring is the same for any N ≥ 2 and the ring is Gorenstein with socle in degree 16. When N = 2, we obtain the Chow ring for the moduli space of elliptic K3 surfaces, we conclude that the Chow ring in this case is tautological.Finally, we present localization computations on the relative Quot scheme over the moduli space of elliptic K3 surfaces. Our calculations are sufficient to determine the divisorial κ-classes in terms of the Hodge class. We also represent one Noether-Lefschetz divisor in terms of the Hodge class, which agrees with the modularity nature of the Noether-Lefschetz divisors.

ISBN: 9798383219294Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Intersection theory

Intersection Theory of the Moduli Space of Elliptic K3 Surfaces.
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300 $a 135 p.
500 $a Source: Dissertations Abstracts International, Volume: 86-01, Section: B.
500 $a Advisor: Oprea, Dragos.
502 $a Thesis (Ph.D.)--University of California, San Diego, 2024.
520 $a Moduli spaces of K3 surfaces are fundamental objects in algebraic geometry. Elliptic K3 surfaces are K3 surfaces with elliptic fibration structure, and they are of particular interest due to their rich geometry. The moduli space of elliptic K3 surfaces can be studied using the theory of Weierstrass models. In this dissertation, we study the topology and intersection theory of the moduli space of elliptic K3 surfaces.We compute the Poincare polynomial of the moduli space of elliptic K3 surfaces. The main idea is constructing a compactification using the Weierstrass models, this compactification is a GIT quotient. We adapt Kirwan's blowup machinery to weighted projective space to compute the Poincare polynomial. We find the cohomology is mostly concentrated in the even degrees, but there is one odd degree class in degree 33.We also study the Chow ring of the moduli space of elliptic surfaces of degree N ≥ 2. We conclude that the Chow ring of the moduli space of elliptic surfaces is always generated by two classes. Furthermore, explicit relations between these classes are given, the Poincare polynomial for the Chow ring is the same for any N ≥ 2 and the ring is Gorenstein with socle in degree 16. When N = 2, we obtain the Chow ring for the moduli space of elliptic K3 surfaces, we conclude that the Chow ring in this case is tautological.Finally, we present localization computations on the relative Quot scheme over the moduli space of elliptic K3 surfaces. Our calculations are sufficient to determine the divisorial κ-classes in terms of the Hodge class. We also represent one Noether-Lefschetz divisor in terms of the Hodge class, which agrees with the modularity nature of the Noether-Lefschetz divisors.
590 $a School code: 0033.
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
653 $a Intersection theory
653 $a Moduli spaces
653 $a Poincare polynomial
653 $a Weierstrass models
653 $a Cohomology
690 $a 0405
690 $a 0642
710 2 $a University of California, San Diego. $b Mathematics. $3 1022364
773 0 $t Dissertations Abstracts International $g 86-01B.
790 $a 0033
791 $a Ph.D.
792 $a 2024
793 $a English
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31300202