## Algebraic Geometry Seminar

Date, time & place Tuesday 10:30 - 11:30 or 12:00 ハイブリッド開催/002Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.)

### 2010/07/05

16:40-18:10   Room #126 (Graduate School of Math. Sci. Bldg.)
Katsuhisa Furukawa (Waseda University)
Rational curves on hypersurfaces (JAPANESE)
[ Abstract ]
Our purpose is to study the family of smooth rational curves of degree $e$ lying on a hypersurface of degree $d$ in $\\mathbb{P}^n$, and to investigate properties of this family (e.g., dimension, smoothness, connectedness).
Our starting point is the research about the family of lines (i.e., $e = 1$), which was studied by W. Barth and A. Van de Ven over $\\mathbb{C}$, and by J. Koll\\'{a}r over an algebraically closed field of arbitrary characteristic.
For the degree $e > 1$, the family of rational curves was studied by J. Harris, M. Roth, and J. Starr over $\\mathbb{C}$ in the case of $d < (n+1)/2$.
In this talk, we study the family of rational curves in arbitrary characteristic under the assumption $e = 2,3$ and $d > 1$, or $e > 3$ and $d > 2e-4$.