# Linear system of divisors

A **linear system of divisors** is a family of effective, linearly equivalent divisors on a non-singular algebraic variety, parametrized by a projective space.<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup> It is one of the central organizing notions of algebraic geometry: it packages the divisors in a fixed linear equivalence class into a single geometric object, and it is the mechanism by which varieties are mapped into projective space.<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup><sup> • </sup><sup>[2](https://poisson.phc.dm.unipi.it/~fpmaiale/notes/GAC.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | A family of effective linearly equivalent divisors on an algebraic variety, parametrized by projective space<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup> |
| Complete system | For an invertible sheaf \(\mathcal{L}\) with finite-dimensional global sections, the complete system \(\lvert \Gamma(X,\mathcal{L}) \rvert\) is the projective space of the section space<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup> |
| Curve case | The complete system \(\lvert D \rvert\) of a divisor \(D\) on a compact Riemann surface is \(\{E \ge 0 : E \sim D\}\), in bijection with \(\mathbb{P}(L(D))\)<sup>[2](https://poisson.phc.dm.unipi.it/~fpmaiale/notes/GAC.pdf)</sup> |
| Dimension on curves | \(\dim \lvert D \rvert = l(D) - 1\), where \(l(D)\) is computed by the Riemann–Roch theorem \(l(D) - l(K-D) = \deg(D) - g + 1\)<sup>[6](https://encyclopediaofmath.org/wiki/Algebraic_curve)</sup> |
| Associated map | A basis \(s_0,\dots,s_n\) of \(L\) defines a rational map \(\phi_L : X \to \mathbb{P}^n\), \(x \mapsto (s_0(x),\dots,s_n(x))\)<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup> |
| Base locus | The locus where the map is undefined; Bertini's theorem says the generic member is smooth away from it<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup><sup> • </sup><sup>[7](https://publications.ias.edu/sites/default/files/alggeom79.pdf)</sup> |
| Classical terminology | A one-dimensional family is a pencil, a two-dimensional one a net, a three- or higher-dimensional one a web<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup> |

## Definition

Let \(X\) be a non-singular algebraic variety over a field \(k\), let \(\mathcal{L}\) be an invertible sheaf on \(X\), and let \(L \subset \Gamma(X,\mathcal{L})\) be a finite-dimensional subspace of the global sections. If \(\dim L > 0\), the divisors determined by the zero loci of sections of \(L\) are effective and linearly equivalent to one another. The linear system is the projective space \(\lvert L \rvert = \mathbb{P}(L)\) of one-dimensional subspaces of \(L\), each line corresponding to one divisor of the family, since scaling a section does not change its zero locus.<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup>

When the full section space \(\Gamma(X,\mathcal{L})\) is finite dimensional, the system \(\lvert \Gamma(X,\mathcal{L}) \rvert\) is called **complete**. Equivalently, for a divisor \(D\), the complete linear system \(\lvert D \rvert\) is the set of all effective divisors \(E \ge 0\) linearly equivalent to \(D\).<sup>[2](https://poisson.phc.dm.unipi.it/~fpmaiale/notes/GAC.pdf)</sup> On a compact [Riemann surface](https://www.edgepedia.org/riemann-surface), the map sending the span of a function \(f \in L(D)\) to the divisor \(\operatorname{div}(f) + D\) is a bijection \(\mathbb{P}(L(D)) \to \lvert D \rvert\), so complete linear systems carry a natural projective space structure.<sup>[2](https://poisson.phc.dm.unipi.it/~fpmaiale/notes/GAC.pdf)</sup> In the modern formulation, a complete linear system of Cartier divisors is viewed through the line bundle \(\mathcal{O}(D)\), and the system is the projectivization of the space of global sections.<sup>[3](https://stacks.math.columbia.edu/tag/0CPG)</sup>

A general (not necessarily complete) linear system is a projective linear subspace inside some complete system: a set of effective divisors, all linearly equivalent, corresponding to a linear subspace of \(\mathbb{P}H^0(X,\mathcal{O}(D))\).<sup>[4](https://math.bu.edu/people/drhast/Notes/Math631-algebraic-geometry-notes.pdf)</sup><sup> • </sup><sup>[5](https://people.sissa.it/~bruzzo/notes/IATG/notes.pdf)</sup>

## Maps to projective space

Linear systems exist because of the connection with projective embeddings. A basis \(s_0, \dots, s_n\) of \(L\) defines a [rational mapping](https://www.edgepedia.org/rational-mapping) \(\phi_L : X \to \mathbb{P}^n\) by \(x \mapsto (s_0(x), \dots, s_n(x))\); one says \(\phi_L\) is defined by the linear system \(\lvert L \rvert\). The image does not lie in any hyperplane, and conversely every rational mapping to projective space with that nondegeneracy property is defined by some linear system.<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup> In geometric terms, the divisors of the system are the preimages of hyperplanes in \(\mathbb{P}^r\) under the associated map.<sup>[8](https://www.ams.org/bookstore/pspdf/gsm-250-prev.pdf)</sup>

This correspondence underlies the classical problem of embedding a variety into projective space: choosing a linear system amounts to choosing the equations that define the map.<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup> The correspondence can be stated precisely: for every scheme \(X\), morphisms \(\phi : X \to \mathbb{P}^r\) with nondegenerate image, taken up to the action of \(\mathrm{PGL}_{r+1}\), correspond naturally and one-to-one with \(r\)-dimensional base-point-free linear series on \(X\), considered up to isomorphism.<sup>[8](https://www.ams.org/bookstore/pspdf/gsm-250-prev.pdf)</sup>

## Base locus and fixed components

A point lies in the base locus of a system when it belongs to the support of every divisor in the family; equivalently, it is where all the defining sections vanish simultaneously, and it is exactly where the associated rational map fails to be defined.<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup><sup> • </sup><sup>[9](https://calwoo.github.io/notes/concepts/algebraic-geometry/note/index.html)</sup> A **fixed component** is an effective divisor \(D^*\) contained in every member, so that each \(D \in \lvert L \rvert\) decomposes as \(D = D' + D^*\). Removing the fixed component leaves a system \(\lvert L' \rvert\) of the same dimension, and the associated map is unchanged, so in studying \(\phi_L\) one may assume the system has no fixed components.<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup> On a curve, subtracting the base divisor from every member of a linear series yields a base-point-free series of the same dimension and lower degree.<sup>[8](https://www.ams.org/bookstore/pspdf/gsm-250-prev.pdf)</sup>

**Bertini's theorem.** The generic member of a linear system is smooth away from the base locus.<sup>[7](https://publications.ias.edu/sites/default/files/alggeom79.pdf)</sup><sup> • </sup><sup>[5](https://people.sissa.it/~bruzzo/notes/IATG/notes.pdf)</sup> This is one of the fundamental theorems of algebraic geometry: it guarantees that the singularities of a general divisor in a system are confined to the part common to all members, and it is the standard tool for constructing smooth varieties by cutting with general members of ample systems.<sup>[10](https://opencw.aprende.org/courses/mathematics/18-725-algebraic-geometry-fall-2015/lecture-notes/MIT18_725F15_lec22.pdf)</sup>

## Examples

- On \(X = \mathbb{P}^2\), the sections of \(\mathcal{O}_{\mathbb{P}^2}(d)\) identify with homogeneous forms of degree \(d\), so the complete linear system \(\lvert \mathcal{O}_{\mathbb{P}^2}(d) \rvert\) is the set of all plane curves of order \(d\).<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup>
- The standard quadratic (Cremona) transformation \(\tau : \mathbb{P}^2 \to \mathbb{P}^2\) is defined by the linear system of conics through the three coordinate points \((0,0,1)\), \((0,1,0)\), and \((1,0,0)\).<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup>
- The Geiser involution is defined by curves of order 8 passing with multiplicity 3 through 7 points in general position; the Bertini involution by curves of order 17 with multiplicity 6 through 8 general points.<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup>
- On \(\mathbb{P}^1(\mathbb{C})\) with \(D = 1 \cdot p\), the system \(\lvert D \rvert\) consists of all divisors \(1 \cdot q\) for \(q \in \mathbb{P}^1\), a one-dimensional family parametrizing the points of the line itself.<sup>[2](https://poisson.phc.dm.unipi.it/~fpmaiale/notes/GAC.pdf)</sup>

In classical projective and analytic geometry one speaks of linear systems of curves, surfaces, or quadrics, meaning families \(\lambda_1 F_1 + \cdots + \lambda_m F_m = 0\). Dimension one gives a pencil, dimension two a net, dimension three or more a web.<sup>[1](https://encyclopediaofmath.org/wiki/Linear_system)</sup>

## Linear systems on curves and Riemann–Roch

On a smooth projective curve, all divisors in a linear system share the same degree, called the degree of the system; a system of degree \(d\) and dimension \(r\) is classically written \(g^r_d\), the "g" standing for *gruppi di punti*, Italian for "divisors".<sup>[8](https://www.ams.org/bookstore/pspdf/gsm-250-prev.pdf)</sup> The dimension of the complete system \(\lvert D \rvert\) is \(l(D) - 1\), where \(l(D)\) is the dimension of the space of functions \(f\) with \((f) + D \ge 0\). Computing \(l(D)\) is a central task of curve theory, governed by the [Riemann–Roch theorem](https://www.edgepedia.org/riemann-roch-theorem) \(l(D) - l(K - D) = \deg(D) - g + 1\), where \(g\) is the genus and \(K\) the canonical divisor of degree \(2g-2\). A divisor with \(l(K-D) > 0\) is called special; divisors of degree greater than \(2g-2\) are non-special, and for them \(l(D) = \deg(D) - g + 1\).<sup>[6](https://encyclopediaofmath.org/wiki/Algebraic_curve)</sup>

The study of linear series on curves and surfaces remains an active area; [Robert Lazarsfeld](https://www.edgepedia.org/robert-lazarsfeld), professor of mathematics, devoted a widely used set of lecture notes to the modern cohomological techniques, centered on Reider's theorem for surfaces.<sup>[11](https://arxiv.org/abs/alg-geom/9408011)</sup>

## Modern extensions

The Cartier divisor formalism makes the section-theoretic picture precise: a regular global section \(s\) of an invertible sheaf \(\mathcal{L}\) has a zero scheme that is an effective Cartier divisor, and the constructions \(D \mapsto (\mathcal{O}_X(D), 1_D)\) and \((\mathcal{L}, s) \mapsto Z(s)\) give mutually inverse bijections between effective Cartier divisors and pairs consisting of an invertible sheaf and a regular global section.<sup>[3](https://stacks.math.columbia.edu/tag/0CPG)</sup>

The notion also extends beyond classical varieties. On a tropical curve, a metric graph with tropical rational functions given by continuous piecewise linear functions of integer slope, the complete linear system \(\lvert D \rvert\) of a divisor \(D\) is defined analogously to the classical one; it carries the structure of a cell complex, linear systems are quotients of finitely generated tropical modules, and \(\lvert D \rvert\) realizes a map from the curve to a tropical projective space whose convex hull is a polyhedral complex.<sup>[12](https://arxiv.org/abs/0909.3685)</sup>

## References

1. [Linear system, Encyclopedia of Mathematics (originator V.A. Iskovskikh)](https://encyclopediaofmath.org/wiki/Linear_system)
2. [F. Maiale, notes on Algebraic Geometry, Chapter 7: Linear systems of divisors, Università di Pisa](https://poisson.phc.dm.unipi.it/~fpmaiale/notes/GAC.pdf)
3. [Section 71.7: Effective Cartier divisors and invertible sheaves, The Stacks Project](https://stacks.math.columbia.edu/tag/0CPG)
4. [D. Hasty, Math 631 Algebraic Geometry notes, Definition 19.20, Boston University](https://math.bu.edu/people/drhast/Notes/Math631-algebraic-geometry-notes.pdf)
5. [A. Bruzzo, notes on algebraic geometry, Chapter 7: Divisors](https://people.sissa.it/~bruzzo/notes/IATG/notes.pdf)
6. [Algebraic curve, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Algebraic_curve)
7. [Algebraic geometry and local differential geometry, IAS lecture notes (Appendix A, Bertini's Theorem)](https://publications.ias.edu/sites/default/files/alggeom79.pdf)
8. [Eisenbud and Harris, The Geometry of Schemes (GSM 250), preview: Linear series and maps to projective space, AMS](https://www.ams.org/bookstore/pspdf/gsm-250-prev.pdf)
9. [Calvin Woo, Classical Algebraic Geometry reference note: Complete Linear System](https://calwoo.github.io/notes/concepts/algebraic-geometry/note/index.html)
10. [MIT 18.725 Algebraic Geometry, Lecture 22: Bertini's Theorem](https://opencw.aprende.org/courses/mathematics/18-725-algebraic-geometry-fall-2015/lecture-notes/MIT18_725F15_lec22.pdf)
11. [R. Lazarsfeld, Lectures on Linear Series, arXiv:alg-geom/9408011](https://arxiv.org/abs/alg-geom/9408011)
12. [C. Haase, G. Musiker, J. Yu, Linear Systems on Tropical Curves, arXiv:0909.3685](https://arxiv.org/abs/0909.3685)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Algebraic geometry › Divisors, cycles, and motives › Divisors, line bundles, and Picard groups*

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