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  "body": "\n## Definition\n\nLet $X$ be a normal [[projective variety|A4H4CZ2V35]] over an algebraically closed field $k$, and let $D$ be a [[Cartier divisor|FHV05ZK99F]] on $X$.\nTwo Cartier divisors are *linearly equivalent* if their difference is principal.\nThe *complete linear system* $\\abs{D}$ is the set of effective Cartier divisors on $X$ that are linearly equivalent to $D$.\nLet $V = H^0(X, \\mathcal{O}_X(D))$, a finite-dimensional vector space, where $\\mathcal{O}_X(D)$ is the [[invertible sheaf|7JCF2TE3V5]] of $D$.\nEach nonzero $s \\in V$ has a divisor of zeros $(s)_0 \\in \\abs{D}$, and $s \\mapsto (s)_0$ gives a bijection between $(V \\setminus \\set{0})/k^*$ and $\\abs{D}$.\nA *linear system* is a subset of $\\abs{D}$ of the form $\\set{(s)_0 : s \\in W \\setminus \\set{0}}$ for a linear subspace $W \\subseteq V$.\nIts dimension is $\\dim W - 1$.\n\n## Remarks\n\nThe *base locus* of a linear system is the intersection of the supports of all its divisors; the system is *base-point-free* if the base locus is empty.\n\nExample: on $X = \\mathbb{P}^n_k$ with $D$ a hyperplane, $V$ is the space of linear forms in $n+1$ variables.\nSo $\\abs{D}$ is the set of all hyperplanes, a linear system of dimension $n$, and it is base-point-free.\n\nNon-example: on $\\mathbb{P}^1_k$, the set of divisors $\\set{p, \\; p + q}$ for two points $p, q$ is not a linear system.\nThe two divisors have degrees $1$ and $2$, so they are not linearly equivalent and do not lie in one $\\abs{D}$.\n\nA base-point-free linear system of dimension $r$, with a basis $s_0, \\dots, s_r$ of $W$, gives a morphism $X \\to \\mathbb{P}^r_k$, $x \\mapsto [s_0(x) : \\dots : s_r(x)]$.\n",
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