Algebraic curve
In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables. More generally an algebraic curve is similar but may be embedded in a higher dimensional space or defined over some more general field.
For example, the unit circle is a real algebraic curve, being the set of zeros of the polynomial x^{2} + y^{2} – 1.
Various technical considerations result in the complex zeros of a polynomial being considered as belonging to the curve. Also, the notion of algebraic curve has been generalized to allow the coefficients of the defining polynomial and the coordinates of the points of the curve to belong to any field, leading to the following definition.
In algebraic geometry, a plane affine algebraic curve defined over a field k is the set of points of K^{2} whose coordinates are zeros of some bivariate polynomial with coefficients in k, where K is some algebraically closed extension of k. The points of the curve with coordinates in k are the k-points of the curve and, all together, are the k part of the curve.
For example, (2,√−3) is a point of the curve defined by x^{2} + y^{2} − 1 = 0 and the usual unit circle is the real part of this curve. The term "unit circle" may refer to all the complex points as well as to only the real points, the exact meaning usually being clear from the context. The equation x^{2} + y^{2} + 1 = 0 defines an algebraic curve, whose real part is empty.
More generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve that is not contained in some plane is called a skew curve. The simplest example of a skew algebraic curve is the twisted cubic. One may also consider algebraic curves contained in the projective space and even algebraic curves that are defined independently to any embedding in an affine or projective space. This leads to the most general definition of an algebraic curve:
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one.
Contents
In Euclidean geometry
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0. This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
With a curve given by such an implicit equation, the first problems are to determine the shape of the curve and to draw it. These problems are not as easy to solve as in the case of the graph of a function, for which y may easily be computed for various values of x. The fact that the defining equation is a polynomial implies that the curve has some structural properties that may help in solving these problems.
Every algebraic curve may be uniquely decomposed into a finite number of smooth monotone arcs (also called branches) sometimes connected by some points sometimes called "remarkable points", and possibly a finite number of isolated points called acnodes. A smooth monotone arc is the graph of a smooth function which is defined and monotone on an open interval of the x-axis. In each direction, an arc is either unbounded (usually called an infinite arc) or has an end point which is either a singular point (this will be defined below) or a point with a tangent parallel to one of the coordinate axes.
For example, for the Tschirnhausen cubic, there are two infinite arcs having the origin (0,0) as end point. This point is the only singular point of the curve. There are also two arcs having this singular point as one end point and having a second end point with a horizontal tangent. Finally, there are two other arcs each having one of these points with horizontal tangent as the first end point and having the unique point with vertical tangent as the second end point. In contrast, the sinusoid is certainly not an algebraic curve, having an infinite number of monotone arcs.
To draw an algebraic curve, it is important to know the remarkable points and their tangents, the infinite branches and their asymptotes (if any) and the way in which the arcs connect them. It is also useful to consider the inflection points as remarkable points. When all this information is drawn on a sheet of paper, the shape of the curve usually appears rather clearly. If not, it suffices to add a few other points and their tangents to get a good description of the curve.
The methods for computing the remarkable points and their tangents are described below, after the section Projective curves.
Plane projective curves
It is often desirable to consider curves in the projective space. An algebraic curve in the projective plane or plane projective curve is the set of the points in a projective plane whose projective coordinates are zeros of a homogeneous polynomial in three variables P(x, y, z).
Every affine algebraic curve of equation p(x, y) = 0 may be completed into the projective curve of equation where
is the result of the homogenization of p. Conversely, if P(x, y, z) = 0 is the homogeneous equation of a projective curve, then P(x, y, 1) = 0 is the equation of an affine curve, which consists of the points of the projective curve whose third projective coordinate is not zero. These two operations are reciprocal one to the other, as and, if p is defined by , then , as soon as the homogeneous polynomial P is not divisible by z.
For example, the projective curve of equation x^{2} + y^{2} − z^{2} is the projective completion of the unit circle of equation x^{2} + y^{2} − 1 = 0.
This implies that an affine curve and its projective completion are the same curve, or, more precisely that the affine curve is a part of the projective curve that is large enough to well define the "complete" curve. This point of view is commonly expressed by calling "points at infinity" of the affine curve the points (in finite number) of the projective completion that do not belong to the affine part.
Projective curves are frequently studied for themselves. They are also useful for the study of affine curves. For example, if p(x, y) is the polynomial defining an affine curve, beside the partial derivatives and , it is useful to consider the derivative at infinity
For example, the equation of the tangent of the affine curve of equation p(x, y) = 0 at a point (a, b) is
Remarkable points of a plane curve
In this section, we consider a plane algebraic curve defined by a bivariate polynomial p(x, y) and its projective completion, defined by the homogenization of p.
Intersection with a line
Knowing the points of intersection of a curve with a given line is frequently useful. The intersection with the axes of coordinates and the asymptotes are useful to draw the curve. Intersecting with a line parallel to the axes allows one to find at least a point in each branch of the curve. If an efficient root-finding algorithm is available, this allows to draw the curve by plotting the intersection point with all the lines parallel to the y-axis and passing through each pixel on the x-axis.
If the polynomial defining the curve has degree d, any line cuts the curve in at most d points. Bézout's theorem asserts that this number is exactly d, if the points are searched in the projective plane over an algebraically closed field (for example the complex numbers), and counted with their multiplicity. The method of computation that follows proves again this theorem, in this simple case.
To compute the intersection of the curve defined by the polynomial p with the line of equation ax+by+c = 0, one solves the equation of the line for x (or for y if a = 0). Substituting the result in p, one gets a univariate equation q(y) = 0 (or q(x) = 0, if the equation of the line has been solved in y), each of whose roots is one coordinate of an intersection point. The other coordinate is deduced from the equation of the line. The multiplicity of an intersection point is the multiplicity of the corresponding root. There is an intersection point at infinity if the degree of q is lower than the degree of p; the multiplicity of such an intersection point at infinity is the difference of the degrees of p and q.
Tangent at a point
The tangent at a point (a, b) of the curve is the line of equation , like for every differentiable curve defined by an implicit equation. In the case of polynomials, another formula for the tangent has a simpler constant term and is more symmetric:
where is the derivative at infinity. The equivalence of the two equations results from Euler's homogeneous function theorem applied to P.
If the tangent is not defined and the point is a singular point.
This extends immediately to the projective case: The equation of the tangent of at the point of projective coordinates (a:b:c) of the projective curve of equation P(x, y, z) = 0 is
and the points of the curves that are singular are the points such that
(The condition P(a, b, c) = 0 is implied by these conditions, by Euler's homogeneous function theorem.)
Asymptotes
Every infinite branch of an algebraic curve corresponds to a point at infinity on the curve, that is a point of the projective completion of the curve that does not belongs to its affine part. The corresponding asymptote is the tangent of the curve at that point. The general formula for a tangent to a projective curve may apply, but it is worth to make it explicit in this case.
Let be the decomposition of the polynomial defining the curve into its homogeneous parts, where p_{i} is the sum of the monomials of p of degree i. It follows that
and
A point at infinity of the curve is a zero of p of the form (a, b, 0). Equivalently, (a, b) is a zero of p_{d}. The fundamental theorem of algebra implies that, over an algebraically closed field (typically, the field of complex numbers), p_{d} factors into a product of linear factors. Each factor defines a point at infinity on the curve: if bx − ay is such a factor, then it defines the point at infinity (a, b, 0). Over the reals, p_{d} factors into linear and quadratic factors. The irreducible quadratic factors define non-real points at infinity, and the real points are given by the linear factors. If (a, b, 0) is a point at infinity of the curve, one says that (a, b) is an asymptotic direction. Setting q = p_{d} the equation of the corresponding asymptote is
If and the asymptote is the line at infinity, and, in the real case, the curve has a branch that looks like a parabola. In this case one says that the curve has a parabolic branch. If
the curve has a singular point at infinity and may have several asymptotes. They may be computed by the method of computing the tangent cone of a singular point.
Singular points
The singular points of a curve of degree d defined by a polynomial p(x,y) of degree d are the solutions of the system of equations:
In characteristic zero, this system is equivalent to
where, with the notation of the preceding section, The systems are equivalent because of Euler's homogeneous function theorem. The latter system has the advantage of having its third polynomial of degree d-1 instead of d.
Similarly, for a projective curve defined by a homogeneous polynomial P(x,y,z) of degree d, the singular points have the solutions of the system
as homogeneous coordinates. (In positive characteristic, the equation has to be added to the system.)
This implies that the number of singular points is finite as long as p(x,y) or P(x,y,z) is square free. Bézout's theorem implies thus that the number of singular points is at most (d−1)^{2}, but this bound is not sharp because the system of equations is overdetermined. If reducible polynomials are allowed, the sharp bound is d(d−1)/2, this value being reached when the polynomial factors in linear factors, that is if the curve is the union of d lines. For irreducible curves and polynomials, the number of singular points is at most (d−1)(d−2)/2, because of the formula expressing the genus in term of the singularities (see below). The maximum is reached by the curves of genus zero whose all singularities have multiplicity two and distinct tangents (see below).
The equation of the tangents at a singular point is given by the nonzero homogeneous part of lowest degree in the Taylor series of the polynomial at the singular point. When one changes the coordinates to put the singular point at the origin, the equation of the tangents at the singular point is thus the nonzero homogeneous part of lowest degree of the polynomial, and the multiplicity of the singular point is the degree of this homogeneous part.
Analytic structure
The study of the analytic structure of an algebraic curve in the neighborhood of a singular point provides an accurate information of the topology of singularities. In fact, near a singular point, a real algebraic curve is the union of a finite number of branches that intersect only at the singular point, and look either as a cusp or as a smooth curve.
Near a regular point, one of the coordinates of the curve may be expressed as an analytic function of the other coordinate. This is a corollary of the analytic implicit function theorem, and implies that the curve is smooth near the point. Near a singular point, the situation is more complicated, and involves Puiseux series, which provide analytic parametric equations of the branches.
For describing a singularity, it is worth to translate the curve for having the singularity at the origin. This consists of a change of variable of the form where are the coordinates of the singular point. In the following, the singular point under consideration is always supposed to be at the origin.
The equation of an algebraic curve is where f is a polynomial in x and y. This polynomial may be considered as a polynomial in y, with coefficients in the algebraically closed field of the Puiseux series in x. Thus f may be factored in factors of the form where P is a Puiseux series. These factors are all different if f is an irreducible polynomial, because this implies that f is square-free, a property which is independent of the field of coefficients.
The Puiseux series that occur here have the form
where d is a positive integer, and is an integer that may also be supposed to be positive, because we consider only the branches of the curve that pass through the origin. Without loss of generality, we may suppose that d is coprime with the greatest common divisor of the n such that (otherwise, one could choose a smaller common denominator for the exponents).
Let be a primitive dth root of unity. If the above Puiseux series occurs in the factorization of , then the d series
occur also in the factorization (a consequence of Galois theory). These d series are said conjugate, and are considered as a single branch of the curve, of ramification index d.
In the case of a real curve, that is a curve defined by a polynomial with real coefficients, three cases may occur. If none has real coefficients, then one has a non-real branch. If some has real coefficients, then one may choose it as . If d is odd, then every real value of x provides a real value of , and one has a real branch that looks regular, although it is singular if d > 1. If d is even, then and have real values, but only for x ≥ 0. In this case, the real branch looks as a cusp (or is a cusp, depending of the definition of a cusp that is used).
For example, the ordinary cusp has only one branch. If it is defined by the equation then the factorization is the ramification index is 2, and the two factors are real and define each an half branch. If the cusp is rotated, it equation becomes and the factorization is with (the coefficient has not been simplified to j for showing how the above definition of is specialized). Here the ramification index is 3, and only one factor is real; this shows that, in the first case, the two factors must be considered as defining the same branch.
Non-plane algebraic curves
An algebraic curve is an algebraic variety of dimension one. This implies that an affine curve in an affine space of dimension n is defined by, at least, n−1 polynomials in n variables. To define a curve, these polynomials must generate a prime ideal of Krull dimension 1. This condition is not easy to test in practice. Therefore, the following way to represent non-plane curves may be preferred.
Let be n polynomials in two variables x_{1} and x_{2} such that f is irreducible. The points in the affine space of dimension n such whose coordinates satisfy the equations and inequations
are all the points of an algebraic curve in which a finite number of points have been removed. This curve is defined by a system of generators of the ideal of the polynomials h such that it exists an integer k such belongs to the ideal generated by . This representation is a rational equivalence between the curve and the plane curve defined by f. Every algebraic curve may be represented in this way. However, a linear change of variables may be needed in order to make almost always injective the projection on the two first variables. When a change of variables is needed, almost every change is convenient, as soon as it is defined over an infinite field.
This representation allows us to deduce easily any property of a non-plane algebraic curve, including its graphical representation, from the corresponding property of its plane projection.
For a curve defined by its implicit equations, above representation of the curve may easily deduced from a Gröbner basis for a block ordering such that the block of the smaller variables is (x_{1}, x_{2}). The polynomial f is the unique polynomial in the base that depends only of x_{1} and x_{2}. The fractions g_{i}/g_{0} are obtained by choosing, for i = 3, ..., n, a polynomial in the basis that is linear in x_{i} and depends only on x_{1}, x_{2} and x_{i}. If these choices are not possible, this means either that the equations define an algebraic set that is not a variety, or that the variety is not of dimension one, or that one must change of coordinates. The latter case occurs when f exists and is unique, and, for i = 3, ..., n, there exist polynomials whose leading monomial depends only on x_{1}, x_{2} and x_{i}.
Algebraic function fields
The study of algebraic curves can be reduced to the study of irreducible algebraic curves: those curves that cannot be written as the union of two smaller curves. Up to birational equivalence, the irreducible curves over a field F are categorically equivalent to algebraic function fields in one variable over F. Such an algebraic function field is a field extension K of F that contains an element x which is transcendental over F, and such that K is a finite algebraic extension of F(x), which is the field of rational functions in the indeterminate x over F.
For example, consider the field C of complex numbers, over which we may define the field C(x) of rational functions in C. If y^{2} = x^{3} − x − 1, then the field C(x, y) is an elliptic function field. The element x is not uniquely determined; the field can also be regarded, for instance, as an extension of C(y). The algebraic curve corresponding to the function field is simply the set of points (x, y) in C^{2} satisfying y^{2} = x^{3} − x − 1.
If the field F is not algebraically closed, the point of view of function fields is a little more general than that of considering the locus of points, since we include, for instance, "curves" with no points on them. For example, if the base field F is the field R of real numbers, then x^{2} + y^{2} = −1 defines an algebraic extension field of R(x), but the corresponding curve considered as a subset of R^{2} has no points. The equation x^{2} + y^{2} = −1 does define an irreducible algebraic curve over R in the scheme sense (an integral, separated one-dimensional schemes of finite type over R). In this sense, the one-to-one correspondence between irreducible algebraic curves over F (up to birational equivalence) and algebraic function fields in one variable over F holds in general.
Two curves can be birationally equivalent (i.e. have isomorphic function fields) without being isomorphic as curves. The situation becomes easier when dealing with nonsingular curves, i.e. those that lack any singularities. Two nonsingular projective curves over a field are isomorphic if and only if their function fields are isomorphic.
Tsen's theorem is about the function field of an algebraic curve over an algebraically closed field.
Complex curves and real surfaces
A complex projective algebraic curve resides in n-dimensional complex projective space CP^{n}. This has complex dimension n, but topological dimension, as a real manifold, 2n, and is compact, connected, and orientable. An algebraic curve over C likewise has topological dimension two; in other words, it is a surface.
The topological genus of this surface, that is the number of handles or donut holes, is equal to the geometric genus of the algebraic curve that may be computed by algebraic means. In short, if one consider a plane projection of a nonsingular curve that has degree d and only ordinary singularities (singularities of multiplicity two with distinct tangents), then the genus is (d − 1)(d − 2)/2 − k, where k is the number of these singularities.
Compact Riemann surfaces
A Riemann surface is a connected complex analytic manifold of one complex dimension, which makes it a connected real manifold of two dimensions. It is compact if it is compact as a topological space.
There is a triple equivalence of categories between the category of smooth irreducible projective algebraic curves over C (with non-constant regular maps as morphisms), the category of compact Riemann surfaces (with non-constant holomorphic maps as morphisms), and the opposite of the category of algebraic function fields in one variable over C (with field homomorphisms that fix C as morphisms). This means that in studying these three subjects we are in a sense studying one and the same thing. It allows complex analytic methods to be used in algebraic geometry, and algebraic-geometric methods in complex analysis, and field-theoretic methods to be used in both. This is characteristic of a much wider class of problems in algebraic geometry.
See also algebraic geometry and analytic geometry for more general theory.
Singularities
Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as smooth (synonymous: non-singular), or else singular. Given n−1 homogeneous polynomials in n+1 variables, we may find the Jacobian matrix as the (n−1)×(n+1) matrix of the partial derivatives. If the rank of this matrix is n−1, then the polynomials define an algebraic curve (otherwise they define an algebraic variety of higher dimension). If the rank remains n−1 when the Jacobian matrix is evaluated at a point P on the curve, then the point is a smooth or regular point; otherwise it is a singular point. In particular, if the curve is a plane projective algebraic curve, defined by a single homogeneous polynomial equation f(x,y,z) = 0, then the singular points are precisely the points P where the rank of the 1×(n+1) matrix is zero, that is, where
Since f is a polynomial, this definition is purely algebraic and makes no assumption about the nature of the field F, which in particular need not be the real or complex numbers. It should, of course, be recalled that (0,0,0) is not a point of the curve and hence not a singular point.
Similarly, for an affine algebraic curve defined by a single polynomial equation f(x,y) = 0, then the singular points are precisely the points P of the curve where the rank of the 1×n Jacobian matrix is zero, that is, where
The singularities of a curve are not birational invariants. However, locating and classifying the singularities of a curve is one way of computing the genus, which is a birational invariant. For this to work, we should consider the curve projectively and require F to be algebraically closed, so that all the singularities which belong to the curve are considered.
Classification of singularities
Singular points include multiple points where the curve crosses over itself, and also various types of cusp, for example that shown by the curve with equation x^{3} = y^{2} at (0,0).
A curve C has at most a finite number of singular points. If it has none, it can be called smooth or non-singular. Commonly, this definition is understood over an algebraically closed field and for a curve C in projective space (i.e., complete in the sense of algebraic geometry). For example, the curve of equation is considered as singular, as having a singular point (a cusp) at infinity.
The singular points are classified by means of several invariants. The multiplicity m is defined as the maximum integer such that the derivatives of f to all orders up to m – 1 vanish (also the minimal intersection number between the curve and a straight line at P). Intuitively, a singular point has delta invariant δ if it concentrates δ ordinary double points at P. To make this precise, the blow up process produces so-called infinitely near points, and summing m(m−1)/2 over the infinitely near points, where m is their multiplicity, produces δ. For an irreducible and reduced curve and a point P we can define δ algebraically as the length of where is the local ring at P and is its integral closure.^{[1]}
The Milnor number μ of a singularity is the degree of the mapping grad f(x,y)/|grad f(x,y)| on the small sphere of radius ε, in the sense of the topological degree of a continuous mapping, where grad f is the (complex) gradient vector field of f. It is related to δ and r by the Milnor-Jung formula,
- μ = 2δ − r + 1.
Here, the branching number r of P is the number of locally irreducible branches at P. For example, r = 1 at an ordinary cusp, and r = 2 at an ordinary double point. Note that m is at least r, and that P is singular if and only if m is at least 2. Moreover, δ is at least m(m-1)/2.
Computing the delta invariants of all of the singularities allows the genus g of the curve to be determined; if d is the degree, then
where the sum is taken over all singular points P of the complex projective plane curve. It is called the genus formula.
Assign the invariants [m, δ, r] to a singularity, where m is the multiplicity, δ is the delta-invariant, and r is the branching number. Then an ordinary cusp is a point with invariants [2,1,1] and an ordinary double point is a point with invariants [2,1,2], and an ordinary m-multiple point is a point with invariants [m, m(m−1)/2, m].
Examples of curves
Rational curves
A rational curve, also called a unicursal curve, is any curve which is birationally equivalent to a line, which we may take to be a projective line; accordingly, we may identify the function field of the curve with the field of rational functions in one indeterminate F(x). If F is algebraically closed, this is equivalent to a curve of genus zero; however, the field of all real algebraic functions defined on the real algebraic variety x^{2}+y^{2} = −1 is a field of genus zero which is not a rational function field.
Concretely, a rational curve embedded in an affine space of dimension n over F can be parameterized (except for isolated exceptional points) by means of n rational functions of a single parameter t; by reducing these rational functions to the same denominator, the n+1 resulting polynomials define a polynomial parametrization of the projective completion of the curve in the projective space. An example is the rational normal curve, where all these polynomials are monomials.
Any conic section defined over F with a rational point in F is a rational curve. It can be parameterized by drawing a line with slope t through the rational point, and intersection with the plane quadratic curve; this gives a polynomial with F-rational coefficients and one F-rational root, hence the other root is F-rational (i.e., belongs to F) also.
For example, consider the ellipse x^{2} + xy + y^{2} = 1, where (−1, 0) is a rational point. Drawing a line with slope t from (−1,0), y = t(x+1), substituting it in the equation of the ellipse, factoring, and solving for x, we obtain
Then the equation for y is
which defines a rational parameterization of the ellipse and hence shows the ellipse is a rational curve. All points of the ellipse are given, except for (−1,1), which corresponds to t = ∞; the entire curve is parameterized therefore by the real projective line.
Such a rational parameterization may be considered in the projective space by equating the first projective coordinates to the numerators of the parameterization and the last one to the common denominator. As the parameter is defined in a projective line, the polynomials in the parameter should be homogenized. For example, the projective parameterization of above ellipse is
Eliminating T and U between these equations we get again the projective equation of the ellipse
which may be easily obtained directly by homogenizing above equation.
Many of the curves on Wikipedia's list of curves are rational, and hence have similar rational parameterizations.
Elliptic curves
An elliptic curve may be defined as any curve of genus one with a rational point: a common model is a nonsingular cubic curve, which suffices to model any genus one curve. In this model the distinguished point is commonly taken to be an inflection point at infinity; this amounts to requiring that the curve can be written in Tate-Weierstrass form, which in its projective version is
If the characteristic of the field is different from 2 and 3, then a linear change of coordinates allows putting which gives the classical Weierstrass form
Elliptic curves carry the structure of an abelian group with the distinguished point as the identity of the group law. In a plane cubic model three points sum to zero in the group if and only if they are collinear. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions.
The intersection of two quadric surfaces is in general a nonsingular curve of genus one and degree four, and thus an elliptic curve, if it has a rational point. In special cases, the intersection either may be a rational singular quartic, or is decomposed in curves of smaller degrees which are not always distinct (either a cubic curve and a line, or two conics, or a conic and two lines, or four lines).
Curves of genus greater than one
Curves of genus greater than one differ markedly from both rational and elliptic curves. Such curves defined over the rational numbers, by Faltings's theorem, can have only a finite number of rational points, and they may be viewed as having a hyperbolic geometry structure. Examples are the hyperelliptic curves, the Klein quartic curve, and the Fermat curve x^{n} + y^{n} = z^{n} when n is greater than three.
See also
Classical algebraic geometry
Modern algebraic geometry
Geometry of Riemann surfaces
Notes
- ^ Hartshorne, Algebraic Geometry, IV Ex. 1.8.
References
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- Egbert Brieskorn and Horst Knörrer, Plane Algebraic Curves, John Stillwell, translator, Birkhäuser, 1986
- Claude Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, American Mathematical Society, Mathematical Surveys Number VI, 1951
- Hershel M. Farkas and Irwin Kra, Riemann Surfaces, Springer, 1980
- W. Fulton, Algebraic Curves: an introduction to algebraic geometry.
- C.G. Gibson, Elementary Geometry of Algebraic Curves: An Undergraduate Introduction, Cambridge University Press, 1998.
- Phillip A. Griffiths, Introduction to Algebraic Curves, Kuniko Weltin, trans., American Mathematical Society, Translation of Mathematical Monographs volume 70, 1985 revision
- Robin Hartshorne, Algebraic Geometry, Springer, 1977
- Shigeru Iitaka, Algebraic Geometry: An Introduction to the Birational Geometry of Algebraic Varieties, Springer, 1982
- John Milnor, Singular Points of Complex Hypersurfaces, Princeton University Press, 1968
- George Salmon, Higher Plane Curves, Third Edition, G. E. Stechert & Co., 1934
- Jean-Pierre Serre, Algebraic Groups and Class Fields, Springer, 1988
- Ernst Kötter (1887). "Grundzüge einer rein geometrischen Theorie der algebraischen ebenen Kurven (Fundamentals of a purely geometrical theory of algebraic plane curves)". Transactions of the Royal Academy of Berlin. — gained the 1886 Academy prize^{[1]}
- ^ Norman Fraser (Feb 1888). "Kötter's synthetic geometry of algebraic curves". Proceedings of the Edinburgh Mathematical Society. 7: 46–61. Here: p.46