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Inmathematics, anaffine algebraic plane curve is thezero set of apolynomial in two variables. Aprojective algebraic plane curve is the zero set in aprojective plane of ahomogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve byhomogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equationh(x,y,t) = 0 can be restricted to the affine algebraic plane curve of equationh(x,y, 1) = 0. These two operations are eachinverse to the other; therefore, the phrasealgebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.
If the defining polynomial of a plane algebraic curve isirreducible, then one has anirreducible plane algebraic curve. Otherwise, the algebraic curve is the union of one or several irreducible curves, called itscomponents, that are defined by the irreducible factors.
More generally, analgebraic curve is analgebraic variety ofdimension one. (In some contexts, an algebraic set of dimension one is also called an algebraic curve, but this will not be the case in this article.) Equivalently, an algebraic curve is an algebraic variety that isbirationally equivalent to an irreducible algebraic plane curve. If the curve is contained in anaffine space or aprojective space, one can take aprojection for such a birational equivalence.
These birational equivalences reduce most of the study of algebraic curves to the study of algebraic plane curves. However, some properties are not kept under birational equivalence and must be studied on non-plane curves. This is, in particular, the case for thedegree andsmoothness. For example, there exist smooth curves ofgenus 0 and degree greater than two, but any plane projection of such curves hassingular points (seeGenus–degree formula).
A non-plane curve is often called aspace curve or askew curve.
An algebraic curve in theEuclidean plane is the set of the points whosecoordinates are the solutions of a bivariatepolynomial equationp(x,y) = 0. This equation is often called theimplicit equation of the curve, in contrast to the curves that are the graph of a function definingexplicitlyy as a function ofx.
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 whichy may easily be computed for various values ofx. 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 monotonearcs (also calledbranches) sometimes connected by some points sometimes called "remarkable points", and possibly a finite number of isolated points calledacnodes. Asmooth monotone arc is the graph of asmooth function which is defined andmonotone on anopen interval of thex-axis. In each direction, an arc is either unbounded (usually called aninfinite arc) or has an endpoint 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 theTschirnhausen cubic, there are two infinite arcs having the origin (0,0) as of endpoint. This point is the onlysingular point of the curve. There are also two arcs having this singular point as one endpoint and having a second endpoint with a horizontal tangent. Finally, there are two other arcs each having one of these points with horizontal tangent as the first endpoint and having the unique point with vertical tangent as the second endpoint. In contrast, thesinusoid 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 theirasymptotes (if any) and the way in which the arcs connect them. It is also useful to consider theinflection 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 in the sectionRemarkable points of a plane curve.
It is often desirable to consider curves in theprojective space. An algebraic curve in theprojective plane orplane projective curve is the set of the points in aprojective plane whoseprojective coordinates are zeros of ahomogeneous polynomial in three variablesP(x,y,z).
Every affine algebraic curve of equationp(x,y) = 0 may be completed into the projective curve of equation whereis the result of thehomogenization ofp. Conversely, ifP(x,y,z) = 0 is the homogeneous equation of a projective curve, thenP(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, ifp is defined by, then as soon as the homogeneous polynomialP is not divisible byz.
For example, the projective curve of equationx2 +y2 −z2 is the projective completion of theunit circle of equationx2 +y2 − 1 = 0.
This implies that an affine curve and its projective completion are the same curves, 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, ifp(x,y) is the polynomial defining an affine curve, beside the partial derivatives and, it is useful to consider thederivative at infinity
For example, the equation of the tangent of the affine curve of equationp(x,y) = 0 at a point (a,b) is
In this section, we consider a plane algebraic curve defined by a bivariate polynomialp(x,y) and its projective completion, defined by the homogenization ofp.
Knowing the points of intersection of a curve with a given line is frequently useful. The intersection with the axes of coordinates and theasymptotes 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 efficientroot-finding algorithm is available, this allows to draw the curve by plotting the intersection point with all the lines parallel to they-axis and passing through eachpixel on thex-axis.
If the polynomial defining the curve has a degreed, any line cuts the curve in at mostd points.Bézout's theorem asserts that this number is exactlyd, if the points are searched in the projective plane over analgebraically closed field (for example thecomplex numbers), and counted with theirmultiplicity. The method of computation that follows proves again this theorem, in this simple case.
To compute the intersection of the curve defined by the polynomialp with the line of equationax+by+c = 0, one solves the equation of the line forx (or fory ifa = 0). Substituting the result inp, one gets a univariate equationq(y) = 0 (orq(x) = 0, if the equation of the line has been solved iny), 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 ofq is lower than the degree ofp; the multiplicity of such an intersection point at infinity is the difference of the degrees ofp andq.
The tangent at a point (a,b) of the curve is the line of equation, like for everydifferentiable 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 fromEuler's homogeneous function theorem applied toP.
If the tangent is not defined and the point is asingular point.
This extends immediately to the projective case: The equation of the tangent of at the point ofprojective coordinates (a:b:c) of the projective curve of equationP(x,y,z) = 0 is
and the points of the curves that are singular are the points such that
(The conditionP(a,b,c) = 0 is implied by these conditions, by Euler's homogeneous function theorem.)
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 belong to its affine part. The correspondingasymptote 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, wherepi is the sum of the monomials ofp of degreei. It follows thatand
A point at infinity of the curve is a zero ofp of the form (a,b, 0). Equivalently, (a,b) is a zero ofpd. Thefundamental theorem of algebra implies that, over an algebraically closed field (typically, the field of complex numbers),pd factors into a product of linear factors. Each factor defines a point at infinity on the curve: ifbx − ay is such a factor, then it defines the point at infinity (a,b, 0). Over the reals,pd factors into linear and quadratic factors. Theirreducible 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 anasymptotic direction. Settingq =pd 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 aparabola. In this case one says that the curve has aparabolic branch. Ifthe 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.
Thesingular points of a curve of degreed defined by a polynomialp(x,y) of degreed are the solutions of the system of equations:Incharacteristic zero, this system is equivalent towhere, with the notation of the preceding section,The systems are equivalent because ofEuler's homogeneous function theorem. The latter system has the advantage of having its third polynomial of degreed-1 instead ofd.
Similarly, for a projective curve defined by a homogeneous polynomialP(x,y,z) of degreed, the singular points have the solutions of the systemashomogeneous 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 asp(x,y) orP(x,y,z) issquare 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 isoverdetermined. Ifreducible polynomials are allowed, the sharp bound isd(d − 1)/2, this value is reached when the polynomial factors in linear factors, that is if the curve is the union ofd 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 the lowest degree in theTaylor 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 the lowest degree of the polynomial, and the multiplicity of the singular point is the degree of this homogeneous part.
The study of theanalytic structure of an algebraic curve in theneighborhood of a singular point provides 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 acusp or as a smooth curve.
Near a regular point, one of the coordinates of the curve may be expressed as ananalytic function of the other coordinate. This is a corollary of the analyticimplicit function theorem, and implies that the curve issmooth near the point. Near a singular point, the situation is more complicated and involvesPuiseux series, which provide analyticparametric equations of the branches.
For describing a singularity, it is worth totranslate 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 wheref is a polynomial inx andy. This polynomial may be considered as a polynomial iny, with coefficients in the algebraically closed field of thePuiseux series inx. Thusf may be factored in factors of the form whereP is a Puiseux series. These factors are all different iff is anirreducible polynomial, because this implies thatf issquare-free, a property which is independent of the field of coefficients.
The Puiseux series that occur here have the formwhered 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 thatd iscoprime with the greatest common divisor of then such that (otherwise, one could choose a smaller common denominator for the exponents).
Let be aprimitivedth root of unity. If the above Puiseux series occurs in the factorization of, then thed seriesoccur also in the factorization (a consequence ofGalois theory). Thesed series are saidconjugate, and are considered as a single branch of the curve, oframification indexd.
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. Ifd is odd, then every real value ofx provides a real value of, and one has a real branch that looks regular, although it is singular ifd > 1. Ifd is even, then and have real values, but only forx ≥ 0. In this case, the real branch looks as acusp (or is a cusp, depending on 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 a half branch. If the cusp is rotated, it equation becomes and the factorization is with (the coefficient has not been simplified toj 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.
An algebraic curve is analgebraic variety ofdimension one. This implies that anaffine curve in anaffine space of dimensionn is defined by, at least,n − 1 polynomials inn variables. To define a curve, these polynomials must generate aprime ideal ofKrull dimension 1. This condition is not easy to test in practice. Therefore, the following way to represent non-plane curves may be preferred.
Let ben polynomials in two variablesx1 andx2 such thatf is irreducible. The points in the affine space of dimensionn 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 polynomialsh such that it exists an integerk such belongs to the ideal generated by.This representation is abirational equivalence between the curve and the plane curve defined byf. 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 theprojection 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 aGröbner basis for ablock ordering such that the block of the smaller variables is (x1,x2). The polynomialf is the unique polynomial in the base that depends only ofx1 andx2. The fractionsgi/g0 are obtained by choosing, fori = 3, ...,n, a polynomial in the basis that is linear inxi and depends only onx1,x2 andxi. If these choices are not possible, this means either that the equations define analgebraic 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 whenf exists and is unique, and, fori = 3, …,n, there exist polynomials whose leading monomial depends only onx1,x2 andxi.
The study of algebraic curves can be reduced to the study ofirreducible algebraic curves: those curves that cannot be written as the union of two smaller curves. Up tobirational equivalence, the irreducible curves over a fieldF arecategorically equivalent toalgebraic function fields in one variable overF. Such an algebraic function field is afield extensionK ofF that contains an elementx which istranscendental overF, and such thatK is a finite algebraic extension ofF(x), which is the field of rational functions in the indeterminatex over F.
For example, consider the fieldC of complex numbers, over which we may define the fieldC(x) of rational functions in C. Ify2 =x3 −x − 1, then the fieldC(x, y) is anelliptic function field. The elementx is not uniquely determined; the field can also be regarded, for instance, as an extension ofC(y). The algebraic curve corresponding to the function field is simply the set of points (x, y) inC2 satisfyingy2 =x3 −x − 1.
If the fieldF 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 fieldF is the fieldR of real numbers, thenx2 +y2 = −1 defines an algebraic extension field ofR(x), but the corresponding curve considered as a subset ofR2 has no points. The equationx2 +y2 = −1 does define an irreducible algebraic curve overR in thescheme sense (anintegral,separatedone-dimensionalschemes offinite type overR). In this sense, the one-to-one correspondence between irreducible algebraic curves overF (up to birational equivalence) and algebraic function fields in one variable overF holds in general.
Two curves can be birationally equivalent (i.e. haveisomorphic function fields) without being isomorphic as curves. The situation becomes easier when dealing withnonsingular 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.
A complex projective algebraic curve resides inn-dimensional complex projective spaceCPn. This has complex dimensionn, but topological dimension, as a realmanifold, 2n, and iscompact,connected, andorientable. An algebraic curve overC likewise has topological dimension two; in other words, it is asurface.
Thetopological genus of this surface, that is the number of handles or donut holes, is equal to thegeometric 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 hasdegreed and only ordinary singularities (singularities of multiplicity two with distinct tangents), then the genus is(d − 1)(d − 2)/2 −k, wherek is the number of these singularities.
ARiemann surface is a connected complex analytic manifold of one complex dimension, which makes it a connected real manifold of two dimensions. It iscompact if it is compact as a topological space.
There is a tripleequivalence of categories between the category of smooth irreducible projective algebraic curves overC (with non-constantregular maps as morphisms), the category of compact Riemann surfaces (with non-constantholomorphic maps as morphisms), and theopposite of the category ofalgebraic function fields in one variable overC (with field homomorphisms that fixC 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 alsoalgebraic geometry and analytic geometry for a more general theory.
Using the intrinsic concept oftangent space, pointsP on an algebraic curveC are classified assmooth (synonymous:non-singular), or elsesingular. Givenn − 1 homogeneous polynomials inn + 1 variables, we may find theJacobian matrix as the (n − 1)×(n + 1) matrix of the partial derivatives. If therank of this matrix isn − 1, then the polynomials define an algebraic curve (otherwise they define an algebraic variety of higher dimension). If the rank remainsn − 1 when the Jacobian matrix is evaluated at a pointP on the curve, then the point is a smooth or regular point; otherwise it is asingular point. In particular, if the curve is a plane projective algebraic curve, defined by a single homogeneous polynomial equationf(x,y,z) = 0, then the singular points are precisely the pointsP where the rank of the 1×(n + 1) matrix is zero, that is, where
Sincef is a polynomial, this definition is purely algebraic and makes no assumption about the nature of the fieldF, 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 equationf(x,y) = 0, then the singular points are precisely the pointsPof 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 thegenus, which is a birational invariant. For this to work, we should consider the curve projectively and requireF to be algebraically closed, so that all the singularities which belong to the curve are considered.
Singular points include multiple points where the curve crosses over itself, and also various types ofcusp, for example that shown by the curve with equationx3 =y2 at (0,0).
A curveC has at most a finite number of singular points. If it has none, it can be calledsmooth ornon-singular. Commonly, this definition is understood over an algebraically closed field and for a curveC in aprojective space (i.e.,complete in the sense of algebraic geometry). For example, the plane curve of equation is considered as singular, as having a singular point (a cusp) at infinity.
In the remainder of this section, one considers a plane curveC defined as the zero set of a bivariate polynomialf(x,y). Some of the results, but not all, may be generalized to non-plane curves.
The singular points are classified by means of several invariants. The multiplicitym is defined as the maximum integer such that the derivatives off to all orders up tom – 1 vanish (also the minimalintersection number between the curve and a straight line atP).Intuitively, a singular point hasdelta invariantδ if it concentratesδ ordinary double points atP. To make this precise, theblow up process produces so-calledinfinitely near points, and summingm(m − 1)/2 over the infinitely near points, wherem is their multiplicity, producesδ.For an irreducible and reduced curve and a pointP we can defineδ algebraically as the length of where is the local ring atP and is its integral closure.[1]
TheMilnor numberμ of a singularity is the degree of the mappinggradf(x,y)/|grad f(x,y)| on the small sphere of radius ε, in the sense of the topologicaldegree of a continuous mapping, wheregrad f is the (complex) gradient vector field off. It is related to δ andr by theMilnor–Jung formula,
Here, the branching numberr ofP is the number of locally irreducible branches atP. For example,r = 1 at an ordinary cusp, andr = 2 at an ordinary double point. The multiplicitym is at leastr, and thatP is singular if and only ifm is at least 2. Moreover, δ is at leastm(m-1)/2.
Computing the delta invariants of all of the singularities allows thegenusg of the curve to be determined; ifd is the degree, then
where the sum is taken over all singular pointsP of the complex projective plane curve. It is called thegenus formula.
Assign the invariants [m, δ,r] to a singularity, wherem is the multiplicity, δ is the delta-invariant, andr is the branching number. Then anordinary cusp is a point with invariants [2,1,1] and anordinary double point is a point with invariants [2,1,2], and an ordinarym-multiple point is a point with invariants [m,m(m − 1)/2,m].
Arational curve, also called a unicursal curve, is any curve which isbirationally 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 indeterminateF(x). IfF is algebraically closed, this is equivalent to a curve ofgenus zero; however, the field of all real algebraic functions defined on the real algebraic varietyx2 + y2 = −1 is a field of genus zero which is not a rational function field.
Concretely, a rational curve embedded in anaffine space of dimensionn overF can be parameterized (except for isolated exceptional points) by means ofnrational functions of a single parametert; by reducing these rational functions to the same denominator, then+1 resulting polynomials define apolynomial parametrization of theprojective completion of the curve in the projective space. An example is therational normal curve, where all these polynomials aremonomials.
Anyconic section defined overF with arational point inF is a rational curve. It can be parameterized by drawing a line with slopet through the rational point, and an intersection with the plane quadratic curve; this gives a polynomial withF-rational coefficients and oneF-rational root, hence the other root isF-rational (i.e., belongs toF) also.
For example, consider the ellipsex2 + xy + y2 = 1, where (−1, 0) is a rational point. Drawing a line with slopet 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 fory 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 tot = ∞; the entire curve is parameterized therefore by the real projective line.
Such a rational parameterization may be considered in theprojective 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 behomogenized. For example, the projective parameterization of the above ellipse is
EliminatingT andU between these equations we get again the projective equation of the ellipsewhich may be easily obtained directly by homogenizing the above equation.
Many of the curves on Wikipedia'slist of curves are rational and hence have similar rational parameterizations.
Rational plane curves are rational curves embedded into. Given generic sections of degree homogeneous polynomials in two coordinates,, there is a map given bydefining a rational plane curve of degree.[2] There is an associatedmoduli space (where is the hyperplane class) parametrizing all suchstable curves. A dimension count can be made to determine the moduli spaces dimension: There are parameters in giving parameters total for each of the sections. Then, since they are considered up to a projective quotient in there is less parameter in. Furthermore, there is a three dimensional group of automorphisms of, hence has dimension. This moduli space can be used to count the number of degree rational plane curves intersecting points usingGromov–Witten theory.[3] It is given by the recursive relationwhere.
Anelliptic curve may be defined as any curve ofgenus one with arational point: a common model is a nonsingularcubic 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 anabelian 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 arecollinear. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo theperiod lattice of the correspondingelliptic functions.
The intersection of twoquadric 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 ofgenus greater than one differ markedly from both rational and elliptic curves. Such curves defined over the rational numbers, byFaltings's theorem, can have only a finite number of rational points, and they may be viewed as having ahyperbolic geometry structure. Examples are thehyperelliptic curves, theKlein quartic curve, and theFermat curvexn +yn =zn whenn is greater than three. Also projective plane curves in and curves in provide many useful examples.
Plane curves of degree, which can be constructed as the vanishing locus of a generic section, have genuswhich can be computed usingCoherent sheaf cohomology. Here's a brief summary of the curves' genera relative to their degree
| degree | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| genus | 0 | 0 | 1 | 3 | 6 | 10 | 15 |
For example, the curve defines a curve of genus which issmooth since the differentials have no common zeros with the curve. A non-example of a generic section is the curve which, byBezout's theorem, should intersect at most points; it is the union of two rational curves intersecting at two points. Note is given by the vanishing locus of and is given by the vanishing locus of. These can be found explicitly: a point lies in both if. So the two solutions are the points such that, which are and.
Curve given by the vanishing locus of, for, give curves of genuswhich can be checked usingCoherent sheaf cohomology. If, then they define curves of genus, hence a curve of any genus can be constructed as a curve in. Their genera can be summarized in the table
| bidegree | ||||
|---|---|---|---|---|
| genus | 1 | 2 | 3 | 4 |
and for, this is
| bidegree | ||||
|---|---|---|---|---|
| genus | 2 | 4 | 6 | 8 |
| Curves |  | ||
|---|---|---|---|
| Helices |
| ||
| Spirals | |||
