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  "body": "\n## Example\n\nLet $k$ be an algebraically closed field and let $m, n \\ge 0$.\nLet $N = (m + 1)(n + 1) - 1$ and give $\\mathbb{P}^N$ the homogeneous coordinates $z_{ij}$ for $0 \\le i \\le m$ and $0 \\le j \\le n$.\nThe *Segre embedding* is the map of [[projective spaces|ZT9A6JK8QF]]\n$$\\sigma \\colon \\mathbb{P}^m \\times \\mathbb{P}^n \\to \\mathbb{P}^N, \\qquad ([x_0 : \\dots : x_m], [y_0 : \\dots : y_n]) \\mapsto [x_i y_j]_{i, j}.$$\n\nCheck that it is well defined: scaling $x$ by $\\lambda$ and $y$ by $\\mu$ scales every $x_i y_j$ by $\\lambda \\mu \\ne 0$, and some $x_i y_j$ is nonzero.\n\nCheck that it is injective: the matrix $(x_i y_j)$ is the outer product $x y^T$, of rank $1$.\nIts nonzero rows are multiples of $y$ and its nonzero columns are multiples of $x$, so the point determines $[x]$ and $[y]$.\n\nThe image is the set of $[z]$ whose $(m+1) \\times (n+1)$ matrix $(z_{ij})$ has rank $1$, that is, the zero set of the [[homogeneous polynomials|2C3PAQ7H50]]\n$$z_{ij} z_{kl} - z_{il} z_{kj}.$$\nThis image is [[irreducible|SY2MWC434Z]], so it is a [[projective variety|A4H4CZ2V35]], and it is used to make $\\mathbb{P}^m \\times \\mathbb{P}^n$ into a projective variety of dimension $m + n$.\n\nFor $m = n = 1$, the image is the quadric surface $z_{00} z_{11} - z_{01} z_{10} = 0$ in $\\mathbb{P}^3$.\nThe fibres $\\sigma(\\set{a} \\times \\mathbb{P}^1)$ and $\\sigma(\\mathbb{P}^1 \\times \\set{b})$ are two families of lines on this surface.\n\n## Explanations\n\n### L1 human\n\nA pair of points, one in each projective space, is turned into one point by multiplying all coordinates of the first with all coordinates of the second.\nThe result is a matrix of rank one, read as a single list of coordinates.\nSo the product of two projective spaces becomes a subset of a bigger projective space, cut out by the vanishing of all $2 \\times 2$ minors.\nFor two projective lines, this gives the familiar saddle-shaped quadric surface, which is covered by two families of straight lines.\n",
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