

Iskovskikh Seminar
November 3, 2016 18:00, Moscow, Steklov Mathematical Institute, room 540






On generating set for cubic hypersurfaces over $\mathbb{Q}$ of high
dimension
Dmitry Mineev^{} ^{} National Research University "Higher School of Economics" (HSE), Moscow

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Abstract:
We will prove that for sufficiently large $n$ on every smooth cubic
hypersurface $X \subset \PP^n (\mathbb{Q})$ there exists a point $P$ which
generates all others through secant and tangent constructions. More
precisely, the following holds: $P = S_0 \subset S_1 \subset \ldots \subset
S_m = X$, where $S_i \setminus S_{i1}$ consists of the points on $X$
through which such a line can be drawn that its other intersections with $X$
lie in $S_{i1}$. We'll need the statement of the theorem by Skinner on weak
approximation. This theorem is the main reason for dimension to be bounded
from below.

