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Hosung Kim, The Space of Rational Curves on a General Hypersurface of Projective Space

April 22, 2021 @ 11:00 am - 12:00 pm KST

B266, IBS Korea, Republic of


Hosung Kim
IBS, Center for Complex Geometry
In 1979, the work of Mori had brought out the importance of the study of rational curves in higher-dimensional geometry. In 1990s, applying Mori’s bend-and-break method, Campana and Kollar-Miyaoka-Mori proved that any Fano manifold is rationally connected. Since then the family of raional curves on Fano maniflolds has been considerably studied especially about the dimension and irreducibility. The expected dimension of the family of rational curves of degree e in a hypersurface of degree d in Pn is e(n-d+1)+n-4, and it has been conjectured that this family is irreducible and has dimension of the expected dimension if d ≤ n-1 and n > 3. This conjecture has been proven for d ≤ n-2 and e arbitrary by Riedl and Yang in 2019 based on bend-and-break. For d = n-1, some results for low e were obtained by Tseng in 2017. In this seminar I am going to introduce a technique based on degenerating the ambient projective space to a 2-component fan and simultaneously degenerate a general hypersurface of a projective space to a subscheme of the fan. Using this method, Ziv Ran reduced the original problem to that on the space of rational curves in Pn which are some secant to a certain (d,d-1) complete intersection, and proved the cases when e < d ≤ n-1 and n > 4.


April 22, 2021
11:00 am - 12:00 pm KST
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IBS Korea, Republic of


Jaehyun Hong
IBS 복소기하학연구단 Center for Complex Geometry
기초과학연구원 복소기하학연구단
대전 유성구 엑스포로 55 (우) 34126
IBS Center for Complex Geometry
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
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