Veronese mapping

$\newcommand{\PP}{\mathbb{P}}$ A special regular mapping of a projective space; named after G. Veronese. Let $n, m$ be positive integers, $v_{nm} = \binom{n+m}{n}-1$, and $\PP^n$, $\PP^{v_{nm}}$ projective spaces over an arbitrary field (or over the ring of integers), regarded as schemes; let $u_0, \ldots, u_n$ be projective coordinates in $\PP^n$, and let $v_{i_0 \cdots i_n}$, $i+0 + \cdots + i_n = m$, be projective coordinates in $\PP^{v_{nm}}$. The Veronese mapping is the morphism

$$v_m : \PP^n \to \PP^{v_{nm}}$$ given by the formulas $v_{i_0 \cdots i_n} = u_0^{i_0} \cdots u_n^{i_n}$, $i_0 + \cdots + i_n = m$. The Veronese mapping may be defined in invariant terms as a regular mapping given by a complete linear system $|mH|$, where $H$ is a hyperplane section in $\PP^n$. The Veronese mapping is a closed imbedding; its image $v_m(\PP^n)$ is called a Veronese variety, and is defined by the equations

$$v_{i_0 \cdots i_n} v_{j_0 \cdots j_n} = v_{k_0 \cdots k_n} v_{r_0 \cdots r_n},$$ where $i_0 + j_0 = k_0 + r_0, \ldots, i_n + j_n = k_n + r_n$. For instance, $v_2(\PP^1)$ is the curve represented by the equation $x_0 x_1 = x_2^2$ in $\PP^2$. The degree of a Veronese variety is $m^n$. For any hypersurface

$$F = \sum_{i_0 + \cdots + i_n = m} a_{i_0 \cdots i_n} u_0^{i_0} \cdots u_n^{i_n} = 0$$ in $\PP^n$ its image with respect to the Veronese mapping $v_m$ is the intersection of the Veronese variety $v_m(\PP^n)$ with the hyperplane

$$\sum_{i_0 + \cdots + i_n = m} a_{i_0 \cdots i_n} v_{i_0 \cdots i_n} = 0.$$ Owing to this fact, Veronese mappings may be used to reduce certain problems on hypersurfaces to the case of hyperplane sections.

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001

Comments

The image of $\PP^2$ in $\PP^5$ under the Veronese imbedding ($n=2$, $m=2$) is called the Veronese surface.

References

[a1]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 178; 674; 179; 349; 525; 532; 535; 632; 743 MR0507725 Zbl 0408.14001
[a2]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381 MR0463157 Zbl 0367.14001