On differential-geometric characteristics of Veronese curves

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 $J_1,J_2,\dots,J_{n-1}$, 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 $J_1\equiv J_2\equiv\dots\equiv J_{n-1}\equiv1$ means that $\gamma$ is a Veronese curve $W^1_n$. On the manifold $\gamma_1\times\gamma_2$ the condition $J_1\equiv J_2\equiv\dots\equiv J_{n-1}\equiv1$ is equivalent to the fact that the curves $\gamma_1$ and $\gamma_2$ lie in one Veronese curve $W^1_n$.