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 Mat. Sb., 2000, Volume 191, Number 7, Pages 73–88 (Mi msb492)

On differential-geometric characteristics of Veronese curves

V. V. Konnov

Moscow State Pedagogical University

Abstract: One part of the algebraizability problem for smooth submanifolds of a projective space is to find differential-geometric invariants of concrete algebraic varieties. In this paper, a property characterizing the Veronese curves $W^1_n$ is discovered and proved. A necessary and sufficient condition for a pair of smooth curves to lie on one Veronese curve is also found. Let $\gamma\times\gamma\setminus\operatorname{diag}(\gamma\times \gamma)$ be the manifold parametrizing pairs of distinct points on a curve $\gamma$, and let $\gamma _1\times \gamma _2$ be the manifold parametrizing pairs of points on two curves $\gamma_1$ and $\gamma_2$ embedded in a projective space $P^n$. A system of differential invariants $J_1,J_2,…,J_{n-1}$, is constructed on the manifolds $\gamma\times \gamma\setminus\operatorname{diag}(\gamma\times\gamma )$ and $\gamma_1\times \gamma_2$. These invariants have the following geometric interpretation. On the manifold $\gamma\times\gamma\setminus\operatorname{diag}(\gamma\times\gamma)$ the condition $J_1\equiv J_2\equiv…\equiv J_{n-1}\equiv1$ means that $\gamma$ is a Veronese curve $W^1_n$. On the manifold $\gamma_1\times\gamma_2$ the condition $J_1\equiv J_2\equiv…\equiv J_{n-1}\equiv1$ is equivalent to the fact that the curves $\gamma_1$ and $\gamma_2$ lie in one Veronese curve $W^1_n$.

DOI: https://doi.org/10.4213/sm492

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English version:
Sbornik: Mathematics, 2000, 191:7, 1015–1031

Bibliographic databases:

UDC: 514.76
MSC: Primary 53A20, 14H45; Secondary 53C10

Citation: V. V. Konnov, “On differential-geometric characteristics of Veronese curves”, Mat. Sb., 191:7 (2000), 73–88; Sb. Math., 191:7 (2000), 1015–1031

Citation in format AMSBIB
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