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  "body": "\n## Example\n\nLet $k$ be an algebraically closed field, and let $n \\ge 1$ and $d \\ge 1$ be integers.\nThere are $\\binom{n + d}{d}$ monomials of degree $d$ in $x_0, \\dots, x_n$; let $N = \\binom{n + d}{d} - 1$.\nThe *Veronese embedding* of degree $d$ is the map of [[projective spaces|ZT9A6JK8QF]]\n$$\\nu_d \\colon \\mathbb{P}^n \\to \\mathbb{P}^N, \\qquad [x_0 : \\dots : x_n] \\mapsto [x^{\\alpha}]_{\\abs{\\alpha} = d},$$\nwhich lists all monomials $x^\\alpha = x_0^{\\alpha_0} \\cdots x_n^{\\alpha_n}$ with $\\alpha_0 + \\dots + \\alpha_n = d$.\n\nCheck that it is a [[morphism|CDS1VWB7A3]]: the monomials are [[homogeneous polynomials|2C3PAQ7H50]] of the same degree $d$.\nSo scaling $x$ by $\\lambda$ scales all of them by $\\lambda^d$.\nThey never vanish together, since $x_i^d$ is among them and some $x_i \\ne 0$.\n\nCheck that it is injective: if $x_i \\ne 0$, then the monomials $x_i^{d-1} x_j$ for $j = 0, \\dots, n$ are the coordinates of $x_i^{d-1} \\cdot (x_0, \\dots, x_n)$, which recover $[x]$.\nIn fact $\\nu_d$ is an isomorphism onto its image, which is a [[projective variety|A4H4CZ2V35]] of dimension $n$.\n\nFor $n = 1$ and $d = 2$, $\\nu_2([s : t]) = [s^2 : st : t^2]$, and the image is the conic $z_0 z_2 = z_1^2$ in $\\mathbb{P}^2$.\nFor $n = 1$ and $d = 3$, the image of $[s : t] \\mapsto [s^3 : s^2 t : s t^2 : t^3]$ is the twisted cubic curve in $\\mathbb{P}^3$.\nFor $n = 2$ and $d = 2$, the image is the Veronese surface in $\\mathbb{P}^5$.\n\nUnder $\\nu_d$, a hypersurface of degree $d$ in $\\mathbb{P}^n$ becomes the intersection of the image with a hyperplane of $\\mathbb{P}^N$.\n\n## Explanations\n\n### L1 human\n\nThe Veronese map replaces a point by the list of all products of its coordinates of a fixed degree.\nFor the projective line and degree two, the point $[s : t]$ goes to $[s^2 : st : t^2]$, and the image is a conic.\nThe trick turns curved equations of degree $d$ into linear equations in the new coordinates.\nThis lets one study hypersurfaces of degree $d$ with the tools of linear algebra on a bigger space.\n",
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