Equivalence problem for minimal rational curves
Classical Algebraic Geometry Today
January 30, 2009 04:00 PM to 05:00 PM
Speakers:
Hwang, Jun-Muk
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Abstract: |
For a family ${\cal K}$ of curves on a variety $X$, a very natural
question is how to describe the configuration of the
members of ${\cal K}$ around a point $x \in X$. This question can
be interpreted as an `equivalence problem' in the sense of E.
Cartan as follows. Given a family ${\cal K}_1$ on $X_1$ and
${\cal K}_2$ on $X_2$ and two points $x_1 \in X_1$ and $x_2 \in
X_2$, when can we find neighborhoods $x_1 \in U_1$ and $x_2 \in
U_2$ and a biholomorphic map $\varphi: U_1 \to U_2$ such that
$\varphi$ (resp. $\varphi^{-1}$) sends pieces of members of ${\cal
K}_1$ (resp. ${\cal K}_2$) to pieces of member of ${\cal K}_2$
(resp. ${\cal K}_1$)? When ${\cal K}_1$ and ${\cal K}_2$ are
minimal rational curves, the problem can be interpreted as the
equivalence problem for certain geometric structures defined by
the tangent vectors of the curves. We will give an overview of
recent development on this problem. |
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Lecture #13565
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