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A special regular mapping of a [[Projective space|projective space]]; named after G. Veronese. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v0966001.png" /> be positive integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v0966002.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v0966003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v0966004.png" /> projective spaces over an arbitrary field (or over the ring of integers), regarded as schemes; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v0966005.png" /> be projective coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v0966006.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v0966007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v0966008.png" />, be projective coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v0966009.png" />. The Veronese mapping is the morphism
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$\newcommand{\PP}{\mathbb{P}}$
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{{TEX|done}}
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A special regular mapping of a
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[[Projective space|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660010.png" /></td> </tr></table>
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$$v_m : \PP^n \to \PP^{v_{nm}}$$
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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
  
given by the formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660012.png" />. The Veronese mapping may be defined in invariant terms as a regular mapping given by a complete linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660014.png" /> is a hyperplane section in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660015.png" />. The Veronese mapping is a closed imbedding; its image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660016.png" /> is called a Veronese variety, and is defined by the equations
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$$v_{i_0 \cdots i_n} v_{j_0 \cdots j_n} = v_{k_0 \cdots k_n} v_{r_0 \cdots r_n},$$
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660017.png" /></td> </tr></table>
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$$F = \sum_{i_0 + \cdots + i_n = m} a_{i_0 \cdots i_n} u_0^{i_0} \cdots u_n^{i_n} = 0$$
 
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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
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660018.png" />. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660019.png" /> is the curve represented by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660021.png" />. The degree of a Veronese variety is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660022.png" />. For any hypersurface
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660023.png" /></td> </tr></table>
 
 
 
in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660024.png" /> its image with respect to the Veronese mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660025.png" /> is the intersection of the Veronese variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660026.png" /> with the hyperplane
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660027.png" /></td> </tr></table>
 
  
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$$\sum_{i_0 + \cdots + i_n = m} a_{i_0 \cdots i_n} v_{i_0 \cdots i_n} = 0.$$
 
Owing to this fact, Veronese mappings may be used to reduce certain problems on hypersurfaces to the case of hyperplane sections.
 
Owing to this fact, Veronese mappings may be used to reduce certain problems on hypersurfaces to the case of hyperplane sections.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich,   "Basic algebraic geometry" , Springer (1977) (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD>
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<TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD>
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</TR></table>
  
  
  
 
====Comments====
 
====Comments====
The image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660029.png" /> under the Veronese imbedding (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096600/v09660031.png" />) is called the Veronese surface.
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The image of $\PP^2$ in $\PP^5$ under the Veronese imbedding ($n=2$, $m=2$) is called the Veronese surface.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 178; 674; 179; 349; 525; 532; 535; 632; 743</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD>
 +
<TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 178; 674; 179; 349; 525; 532; 535; 632; 743 {{MR|0507725}} {{ZBL|0408.14001}} </TD>
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</TR><TR><TD valign="top">[a2]</TD>
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<TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381 {{MR|0463157}} {{ZBL|0367.14001}} </TD>
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</TR></table>

Latest revision as of 23:35, 22 October 2018

$\newcommand{\PP}{\mathbb{P}}$ 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
How to Cite This Entry:
Veronese mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Veronese_mapping&oldid=13814
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article