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  "body": "\n## Summary\n\nA linear system on a projective variety is a family of linearly equivalent effective divisors, parametrized by a projective space.\nThe complete linear system $\\abs{D}$ of a divisor $D$ consists of all effective divisors linearly equivalent to $D$.\nIt is the projectivization of the space of global sections of $\\mathcal{O}_X(D)$.\nA linear system without base points gives a morphism to projective space.\nUp to a choice of coordinates, every morphism to projective space arises in this way.\nThey are the basic tool to embed varieties and to study maps between them.\n\n## Explanations\n\n### L1 human\n\nTake all the curves of a fixed kind on a surface, such as the conics in the plane.\nEach conic is the zero set of a quadratic polynomial, and two polynomials that differ by a nonzero scalar give the same conic.\nSo the conics form a projective space, here of dimension $5$, because a quadratic polynomial in three variables has six coefficients.\nThis family is a linear system: one can add polynomials, so the family is \"linear\".\nThe lines in the plane form a smaller linear system of dimension $2$.\n",
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