In [GH], Griffiths and Harris asked whether a projective complex submanifold of codimension two is determined by the moduli of its second fundamental forms. More precisely, given a nonsingular subvariety $X^n$ ⊂ $P^{n+2}$, the second fundamental form $II_{X,x}$ at a point x ∈ X is a pencil of quadrics on $T_x(X)$, defining a rational map $µ^x$ from X to a suitable moduli space of pencils of quadrics on a complex vector space of dimension n. The question raised by Griffiths and Harris was whether the image of $µ^X$ determines X. We study this question when $X^n$ ⊂ $P^{n+2}$ is a nonsingular intersection of two quadric hypersurfaces of dimension n > 4. In this case, the second fundamental form $II_{X,x}$ at a general point x ∈ X is a nonsingular pencil of quadrics. Firstly, we prove that the moduli map $µ^X$ is dominant over the moduli of nonsingular pencils of quadrics. This gives a negative answer to Griffiths-Harris’s question. To remedy the situation, we consider a refined version $µe^X$ of the moduli map $µ^X$, which takes into account the infinitesimal information of $\widetilde\mu^X$. Our main result is an affirmative answer in terms of the refined moduli map: we prove that the image of $\widetilde\mu^X$ determines X, among nonsingular intersections of two quadrics.

- Advisors
- Kwak, Sijong
*researcher*; 곽시종*researcher*

- Description
- 한국과학기술원 :수리과학과,

- Publisher
- 한국과학기술원

- Issue Date
- 2018

- Identifier
- 325007

- Language
- eng

- Description
학위논문(박사) - 한국과학기술원 : 수리과학과, 2018.2,[i, 53 p. :]

- Keywords
Complete intersections of two quadrics▼aSecond fundamental forms▼aComplex geometry▼aAlgebraic geometry▼aProjective differential geometry; 두 2차초곡면의 교집합▼a2차형식▼a복소기하▼a대수기하▼a사영미분기하

- Appears in Collection
- MA-Theses_Ph.D.(박사논문)

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