Glossary of classical algebraic geometry

The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of the century, and later formalized by André Weil, Jean-Pierre Serre and Alexander Grothendieck. Much of the classical terminology, mainly based on case study, was simply abandoned, with the result that books and papers written before this time can be hard to read. This article lists some of this classical terminology, and describes some of the changes in conventions.

translates many of the classical terms in algebraic geometry into scheme-theoretic terminology. Other books defining some of the classical terminology include, , , , ,.

Conventions
The change in terminology from around 1948 to 1960 is not the only difficulty in understanding classical algebraic geometry. There was also a lot of background knowledge and assumptions, much of which has now changed. This section lists some of these changes.


 * In classical algebraic geometry, adjectives were often used as nouns: for example, "quartic" could also be short for "quartic curve" or "quartic surface".
 * In classical algebraic geometry, all curves, surfaces, varieties, and so on came with fixed embeddings into projective space, whereas in scheme theory they are more often considered as abstract varieties. For example, a Veronese surface was not just a copy of the projective plane, but a copy of the projective plane together with an embedding into projective 5-space.
 * Varieties were often considered only up to birational isomorphism, whereas in scheme theory they are usually considered up to biregular isomorphism.
 * Until circa 1950, many of the proofs in classical algebraic geometry were incomplete (or occasionally just wrong). In particular authors often did not bother to check degenerate cases.
 * Words (such as azygetic or bifid) were sometimes formed from Latin or Greek roots without further explanation, assuming that readers would use their classical education to figure out the meaning.


 * Definitions in classical algebraic geometry were often somewhat vague, and it is futile to try to find the precise meaning of some of the older terms because many of them never had a precise meaning. In practice this did not matter much when the terms were only used to describe particular examples, as in these cases their meaning was usually clear: for example, it was obvious what the 16 tropes of a Kummer surface were, even if "trope" was not precisely defined in general.
 * Algebraic geometry was often implicitly done over the complex numbers (or sometimes the real numbers).
 * Readers were often assumed to know classical (or synthetic) projective geometry, and in particular to have a thorough knowledge of conics, and authors would use terminology from this area without further explanation.
 * Several terms, such as "Abelian group", "complete", "complex", "flat", "harmonic", "homology", "monoid", "normal", "pole", "regular", now have meanings that are unrelated to their original meanings. Other terms, such as "circle", have their meanings tacitly changed to work in complex projective space; for example, a circle in complex algebraic geometry is a conic passing through the circular points at infinity and has underlying topological space a 2-sphere rather than a 1-sphere.
 * Sometimes capital letters are tacitly understood to stand for points, and small letters for lines or curves.

Symbols
[1], [2], . . ., [n]: Projective space of dimension $1, 2, \ldots, n$. This notation was introduced by. ∞¹, ∞², ...: A family of dimension 1, 2, ... {1}, {2}, ...,{n}: A family or variety of dimension $1, 2, \ldots, n$.

A
Abelian group: An archaic name for the symplectic group. A commutative group.

aberrancy: The deviation of a curve from circular form. See.

absolute: A fixed choice of something in projective space, used to construct some other geometry from projective geometry. For example, choosing a plane, called the absolute plane, of projective space can be used to make its complement into a copy of affine space. Choosing a suitable conic or polarity, called the Cayley absolute, absolute conic or absolute polarity, in the absolute plane provides the means to put a metric on affine space so that it becomes a metric space. Absolute geometry is roughly Euclidean geometry without the parallel postulate.

accidental: An accidental (or improper) double point of a surface in 4-dimensional projective space is a double point with two distinct tangent planes.

acnode: An acnode is an isolated point of a real curve. See.

adjoint: If C is a curve, an adjoint of C is a curve such that any point of C of multiplicity r has multiplicity at least r–1 on the adjoint. Sometimes the multiple points of C are required to be ordinary, and if theis condition is not satisfied the term "sub-adjoint" is used.

affine: Affine space is roughly a vector space where one has forgotten which point is the origin. An affine variety is a variety in affine space.

affinity: An automorphism of affine space.

aggregate: A set.

ambient: An ambient variety is a large variety containing all the points, curves, divisors, and so on that one is interested in.

anharmonic ratio: Cross-ratio

antipoint: One of a pair of points constructed from two foci of a curve. See.

apparent: An apparent singularity is a singularity of a projection of a variety into a hyperplane. They are so called because they appear to be singularities to an observer at the point being projected from.

apolar: Orthogonal under the polar pairing between the symmetric algebra of a vector space and its dual.

arithmetic genus: The arithmetic genus of a variety is a variation of the Euler characteristic of the trivial line bundle; see Hodge number.

Aronhold set: One of the 288 sets of 7 of the 28 bitangents of a quartic curve corresponding to the 7 odd theta characteristics of a normal set.

associated: An associated curve is the image of a projective curve in a Grassmannian, given by taking the tangent lines, or osculating planes, and so on.

axial: axis: A special line or linear subspace associated with some family of geometric objects. For example, a special linear complex in 4-dimensional space consists of all lines meeting a given plane, that is called the axial plane of the complex. Similar to directrix.

azygetic: Unpaired. Opposite of syzygetic, meaning paired. Example: azygetic triad, azygetic tetrad, azygetic set.

B
base: A base point is a point common to all members of a family. The base number ρ is the rank of the Neron–Severi group.

bicircular: Having nodes at the two circular points at infinity, as in bicircular curve. See.

bicorn: A bicorn is a curve with two cusps.

bicuspidal: Having two cusps

bidegree: A pair of integers giving the degrees of a bihomogeneous polynomial in two sets of variables

bielliptic: A bielliptic curve is a branched double cover of an elliptic curve. A bielliptic surface is the same as a hyperelliptic surface.

bifid: Split into two equal parts A bifid map is an element of the vector space of dimension 2g over the field with 2 elements, consisting of the 2g+1-dimensional space of even-cardinality subsets of a set S of 2+2g elements, modulo the 1-dimensional space {0,S}. A bifid substitution is a permutation of the 28 bitangents of a quartic curve depending on one of the 35 decompositions of 8 symbols into two sets of 4 symbols. See.

biflecnode: Same as fleflecnode. See.

bigenus: The second plurigenus P2 of a surface.

bihomogeneous: Homogeneous in each of two sets of variables, as in bihomogeneous form.

binary: Depending on two variables, as in binary form

binodal: Having two nodes

binode: A double point of a surface whose tangent cone consists of two different planes. See unode.

bipartite: Having two connected components. See.

bipunctual: Having two points For a bipunctual conic with respect to 3 points see.

birational: Two varieties are birational if they are isomorphic off lower-dimensional subsets A birational map is a rational map with rational "inverse"

biregular: A biregular map is a regular map with regular inverse Two varieties are biregular if there is a biregular map from one to the other, in other words if they are isomorphic as abstract varieties.

biscribed: Both circumscribed and inscribed, or in other words having vertices that lie on a curve and sides that are tangent to the curve, as in biscribed triangle.

bitangent: A bitangent is a line that is tangent to a curve at two points. See.

bitangential: Meeting a curve at the tangency points of its bitangents

Brianchon hexagon: A non-planar hexagon whose three diagonals meet.

C
canonical: The canonical series is the linear series of the canonical line bundle The canonical bundle is the line bundle of differential forms of highest degree. The canonical map or canonical embedding is the map to the projective space of the sections of the canonical bundle A canonical curve (or variety) is the image of a curve (or variety) under the canonical map The canonical class is the divisor class of a canonical divisor A canonical divisor is a divisor of a section of the canonical line bundle.

catalecticant: A catalecticant is an invariant of a binary form of degree 2n that vanishes when the form is a sum of powers of n linear forms.

caustic: A caustic is the envelope of light rays from a point reflected in a curve

Cayley: Cayleyan: Named after Arthur Cayley See A Cayley octad is a set of 8 points in projective space given by the intersection of three quadrics. The Cayley lines or Cayley–Salmon lines are the 20 lines passing through 3 Kirkman points. A Cayley absolute is a conic or quadric used to define a metric.

center: centre: A special point associated with some geometric object The center of a perspectivity The center of an isologue

character: characteristic: An integer associated with a projective variety, such as its degree, rank, order, class, type. In particular the Plücker characteristics of a curve are the order, class, number of nodes, number of bitangents, number of cusps, and number of inflections. A characteristic exponent is an exponent of a power series with non-negative coefficient, that is not divisible by the highest common factor of preceding exponents with non-zero coefficients. The characteristic series of a linear system of divisors on a surface is the linear system of 0-cycles on one of the divisors given by its intersections with the other divisors.

chord: A line joining two points of a variety

chordal variety: A chordal variety is the union of the chords and tangent spaces of a projective variety

circle: A plane conic passing through the circular points at infinity. For real projective geometry this is much the same as a circle in the usual sense, but for complex projective geometry it is different: for example, circles have underlying topological spaces given by a 2-sphere rather than a 1-sphere.

circuit: A component of a real algebraic curve. A circuit is called even or odd depending on whether it has an even or odd number of intersections with a generic line.

circular: A circular point is one of the two points at infinity (1: i: 0), (1: −i: 0) through which all circles pass A circular algebraic curve is a curve passing through the two circular points at infinity. See also bicircular.

circumscribed: Having edges tangent to some curve, as in circumscribed quadrilateral. Passing through the vertices of something, as in circumscribed circle.

cissoid: A cissoid is the curve generated from two curves and a point. See.

class: The class of a plane curve is the number of proper tangents passing through a generic point of the plane. The class of a space curve is the number of osculating planes passing through a generic point of space. The class of a surface in rdimensional projective space is the number of tangent planes meeting a generic codimension 2 subspace in a line. The degree of a contravariant or concomitant in the covariant variables.

coaxal: coaxial: A pencil of circles is called coaxal if their centers all lie on a line (called the axis). A family of plane circles all passing through the same two points (other than the circular points at infinity).

coincidence: A coincidence quadric is a quadric associated to a correlation, given by the locus of points lying in the corresponding hyperplane. A fixed point of a correspondence, in other words a point of a variety corresponding to itself under a correspondence.

collinear: On the same line

collineation: A collineation is an isomorphism from one projective space to another, often to itself. See correlation.

complete: A linear series of divisors is called complete if it is not contained in a larger linear series. A scheme is called complete if the map to a point is proper A complete quadrangle is 4 points and the 6 lines joining pairs A complete quadrilateral is 4 lines meeting in pairs in 6 points A complete conic in the plane is a (possibly degenerate) conic, together with a pair of (possibly equal) points on it if it is a double line

complex: (Noun.) A line complex, a family of lines of codimension 1 in the family of all lines in some projective space, in particular a 3-dimensional family of lines in 3-dimensional projective space. See congruence.

The (line) complex group is an old name for the symplectic group.

composite: Reducible (meaning having more than one irreducible component).

conchoid: A conchoid is the curve given by the cissoid of a circle and another curve. See.

concomitant: A (mixed) concomitant is an invariant homogeneous polynomial in the coefficients of a form, a covariant variable, and a contravariant variable. In other words it is a (tri)homogeneous polynomial on SV&oplus;V&oplus;V* for some vector space V, where SV is some symmetric power of V and V* its dual, that is invariant under the special linear group of V. In practice V often has dimension 2. The degree, class, and order of a concomitant are its degrees in the three types of variable. Concomitants are generalizations of covariants, contravariants, and invariants.

concurrent: Meeting at a point

cone: The union of the lines joining an algebraic set with a linear algebraic set. Called a point-cone, line-cone, ... if the linear set is a point, line, ... A subset of a vector space closed under multiplication by scalars.

configuration: A configuration is a finite set of points and lines (and sometimes planes), generally with equal numbers of points per line and equal numbers of lines per point.

confocal: Having the same foci

congruence: A family of lines in projective space such that there are a nonzero finite number of lines through a generic point. See complex.

conic: A conic is a degree 2 curve. Short for "conic section", the intersection of a cone with a plane.

conjugate: A conjugate point is an acnode. A conjugate point is a point lying on the hyperplane corresponding to another point under a polarity. A conjugate line is a line containing the point corresponding to another line under a polarity (or plane conic). For harmonic conjugate see harmonic.

connex: A correspondence between a projective space and its dual.

consecutive: Infinitesimally near. For example, a tangent line to a curve is a line through two consecutive points of the curve, and a focal point is the intersection of the normals of two consecutive points.

contravariant: A bihomogeneous polynomial in dual variables of x, y, ... and the coefficients of some homogeneous form in x, y,... that is invariant under some group of linear transformations. In other words it is a bihomogeneous polynomial on SV&oplus;V for some vector space V, where SV is some symmetric power of V and V* its dual, that is invariant under the special linear group of V. In practice V often has dimension at least 3, because when it has dimension 2 these are more or less the same as covariants. The degree and class of a contravariant are its degrees in the two types of variable. Contravariants generalize invariants and are special cases of concomitants, and are in some sense dual to covariants.

coplanar: In the same plane

correlation: An isomorphism from a projective space to the dual of a projective space, often to the dual of itself. A correlation on the projective space of a vector space is essentially the same as a nonsingular bilinear form on the vector space, up to multiplication by constants.

coresidual: See

correspondence: A correspondence from X to Y is an algebraic subset of X&times;Y

cosingular: Having the same singularities

couple: An ordered pair

covariant: A bihomogeneous polynomial in x, y, ... and the coefficients of some homogeneous form in x, y,... that is invariant under some group of linear transformations. In other words it is a bihomogeneous polynomial on SV&oplus;V* for some vector space V, where SV is some symmetric power of V and V* its dual, that is invariant under the special linear group of V. In practice V often has dimension 2. The degree and order of a covariant are its degrees in the two types of variable. Covariants generalize invariants and are special cases of concomitants, and are in some sense dual to contravariants The variety defined by a covariant. In particular the curve defined by the Hessian or Steinerian covariants of a curve are called covariant curves.

Cremona transformation: A Cremona transformation is a birational map from a projective space to itself

cross-ratio: The cross-ratio is an invariant of 4 points on a projective line.

crunode: Crunode is an archaic term for a node, a double point with distinct tangent directions.

cubic: Degree 3, especially a degree 3 projective variety

cubo-cubic: A cubo-cubic transformation is a Cremona transformation such that the homaloids of the transformation and its inverse all have degree 3.

curve: A curve together with an embedding into projective space.

cusp: A cusp is a singular point of a curve whose tangent cone is a line.

cuspidal edge: The locus of the focal points of a family of planes

cyclide: A cyclide is a quartic surface passing doubly through the absolute conic.

D
decic: decimic:

deficiency: The deficiency of a linear system is its codimension in the corresponding complete linear system. The deficiency D of a plane curve is an approximation to its genus, equal to the genus when all singular points are ordinary, given by (n–1)(n–2)/2 –(a–1)(a–2)/2 – (b–1)(b–2)/2 –..., where n is the degree of the curve and a. b, ... are the multiplicities of its singular points. ,

degree: The number of intersection points of a projective variety with a generic linear subspace of complementary dimension The number of points of a divisor on a curve

Desargues: The Desargues figure or configuration is a configuration of 10 lines and 10 points in Desargues' theorem.

desmic system: A desmic system is a configuration of three desmic tetrahedra.

developable:

(Noun) Short for a developable surface, one that can be unrolled to a plane

Flat, as in developable surface

differential: A differential of the first kind is a holomorphic 1-form. A differential of the second kind is a meromorphic 1-form such that the residues of all poles are 0. Sometimes it is only allowd to have one pole that must be of order 2. A differential of the third kind is sometimes a meromorphic 1-form such that all poles are simple (order 1). Sometimes it is only allowed to have 2 poles.

director: The director circle of a conic is the locus of points where two orthogonal tangent lines to the conic meet. More generally the director conic of a conic in regard to two points is defined in a similar way.

directrix: A straight line, or more generally a projective space, associated with some geometric configuration, such as the directrix of a conic section or the directrix of a rational normal scroll

discriminant: The invariant (on the vector space of forms of degree d in n variables) which vanishes exactly when the corresponding hypersurface in Pn-1 is singular.

double curve: A 1-dimensional singularity, usually of a surface, of multiplicity 2

double point: A 0-dimensional singularity of multiplicity 2, such as a node. One of the two points fixed by an involution of a projective line.

double six: The Schläfli double six configuration

duad: A set of two points

dual:

A dual number is a number of the form a+εb where ε has square 0.

E
env Eckardt point: An Eckardt point is a point of intersection of 3 lines on a cubic surface.

effective: An effective cycle or divisor is one with no negative coefficients

elation: A collineation that fixes all points on a line (called its axis) and all lines though a point on the axis (called its center).

eleven-point conic: The eleven-point conic is a conic containing 11 special points associated to four points and a line.

embedded: An embedded variety is one contained in a larger variety, sometimes called the ambient variety.

enneaedro: A set of 9 tritangent planes to a cubic surface containing the 27 lines.

envelope: A curve tangent to a family of curves. See.

epitrochoid: An epitrochoid is the curve traced by a point of a disc rolling along another disc.

equiaffine: equiaffinity: An equiaffinity is an equiaffine transformation, meaning an affine transformation preserving area.

equianharmonic: Four points whose cross ratio (or anharmonic ratio) is a cube root of 1 An equianharmonic cubic is a cubic curve with j-invariant 0

equivalence: In intersection theory, a positive-dimensional variety sometimes behaves formally as if it were a finite number of points; this number is called its equivalence.

evectant: A contravariant defined by Sylvester depending on an invariant. See.

evolute: An evolute is the envelope of the normal lines of a plane curve. See.

exceptional: Corresponding to something of lower dimension under a birational correspondence, as in exceptional curve, exceptional divisor An exceptional curve on a surface is one that corresponds to a simple point on another surface under a birational correspondence. It is called an exceptional curve of the first kind if it is transformed into a point of the other surface, and an exceptional curve of the second kind if it is transformed into a curve of the other surface.

F
facultative: A facultative point is one where a given function is positive.

first kind: holomorphic or regular (when applied to differentials)

flat: (Noun) A linear subspace of projective space, such as a point, line, plane, hyperplane. (Adjective) Having curvature zero. (Adjective) For the term "flat" in scheme theory see flat module, flat morphism.

flecnode: A double point that is also a point of inflexion of one branch..

fleflecnode: A double point that is also a point of inflexion of both branches..

flex: Short for point of inflection

focal: A focal point, line, plane, ... is the intersection of several consecutive elements of a family of linear subspaces. A focal curve, surface and so on is the locus of the focal points of a family of linear subspaces.

focus: A focal point. See ,

foliate singularity: See

form: A homogeneous polynomial in several variables. Same as quantic. A differential form.

free intersection: An intersection point of two members of a family that is not a base point.

freedom: Dimension, as in degrees of freedom..

fundamental: This term seem to be ambiguous and poorly defined: Zariski states: "I can find no clear-cut definition of a fundamental curve in the literature". The fundamental set or fundamental locus of a birational correspondence appears to mean (roughly) either the set of points where it is not a bijection or the set of points where it is not defined. A fundamental point, curve, or variety is a point, curve, or variety in the fundamental set of a birational correspondence.

G
g: A linear or algebraic system of divisors of dimension r and degree d on a curve. The letter g is used for linear systems, and the letter γ is used for algebraic systems. generator: One of the lines of a ruled surface or more generally an element of some family of linear spaces.

generic: Not having some special properties, which are usually not stated explicitly. A generic point is one having coordinates that are algebraically independent over the base field. The generic point of a scheme.

genus:

geometric genus: The geometric genus is the dimension of the space of holomorphic n-forms on an n-dimensional non-singular projective variety.

grade: The grade of a linear system of divisors on an n-dimensional variety is the number of free intersection points of n generic divisors. In particular the grade of a linear series of divisors on a curve is now called the degree and is the number of points in each divisor, and the grade of a net of curves on a surface is the number of free intersections of two generic curves.

Grassmannian: A Grassmannian is a variety parameterizing linear subspaces of projective space

group: A group or point-group is an archaic term for an effective divisor on a curve. This usage is particularly confusing, because some such divisors are called normal, with the result that there are "normal sub-groups" having nothing to do with the normal subgroups of group theory. A group in the usual sense.

H
harmonic: Two pairs of points on a line are harmonic if their cross ratio is –1. The 4 points are called a harmonic set, and the points of one pair are called harmonic conjugates with respect to the other pair. A harmonic cubic is an elliptic curve with j-invariant 1728, given by a double cover of the projective line branched at 4 points with cross ratio –1. Satisfying some analogue of the Laplace equation, as in harmonic form. The harmonic polar line of an inflection point of a cubic curve is the component of the polar conic other than the tangent line. A harmonic net is a set of points on a line containing the harmonic conjugate of any point with respect to any other two points. For harmonically conjugate conics see.

Hesse: Hessian: Named after Otto Hesse. A Hessian matrix, or a variety associated with it. See. The Hessian line is a line associated to 3 points A, B, C, of a conic, containing the three points given by the intersections of the tangents at A, B, C with the lines BC, CA, AB. The Hessian point is a point associated to three lines tangent to a conic, whose construction is dual to that of a Hessian line. The Hessian pair or Hessian duad of three points on a projective line is the pair of points fixed by the projective transformations of order 3 permuting the 3 points. More generally the Hessian pair is also defined in a similar way for triples of points of a rational curve, or triples of elements of a pencil. The Hesse configuration is the configuration of inflection points of a plane cubic. The Hesse group is the group of automorphisms of the Hesse configuration, of order 216.

hexad: A set of 6 points

homaloid: An element of a homaloidal system, in particular the image of a hyperlpane under a Cremona transformation. homaloidal: A homaloidal linear system of divisors is a linear system of grade 1, such as the image of the linear system of hyperplanes of projective space under a Cremona transformation. When the linear system has dimension 2 or 3 it is called a homaloidal net or homaloidal web. Homaloidal means similar to a flat plane.

homographic: Having the same invariants. See. A homographic transformation is an automorphism of projective space over a field, in other words an element of the projective general linear group.

homography: An isomorphism between projective spaces induced by an isomorphism of vector spaces. An axis of homography is a line associated to two related ranges of a conic.

homology: As in homology group A collineation fixing all lines through a point (the center) and all points through a line (the axis) not containing the center. See elation. This terminology was introduced by Lie. An automorphism of projective space with a hyperplane of fixed points (called the axis). It is called a harmonic homology if it has order 2, in which case it has an isolated fixed point called its center.

Hurwitz curve: Hurwitz surface: A Hurwitz curve is a complex algebraic curve of genus g>0 with the maximum possible number 84(g–1) of automorphisms.

hyperbolism: Essentially a blow-up of a curve at a point. See.

hypercusp: A singularity of a curve of some multiplicity r whose tangent cone is a single line meeting the curve with order r+1.

hyperelliptic: A hyperelliptic curve is a curve with a degree 2 map to the projective line.

hyperflex: Same as point of undulation: a point of a curve where the tangent line has contact of order at least 4.

hyperosculating point: A point where the tangent space meets with order higher than normal.

hyperplane: A linear subspace of projective space of codimension 1. Same as prime.

I
index of speciality: The dimension of the first cohomology group of the line bundle of a divisor D; often denoted by i or i(D).

infinitely near point: A point on a blow up of a variety

inflection: inflexion: An inflection is a point where the curvature vanishes, or in other words where the tangent line meets with order at least 3. Differential geometry uses the slightly stricter condition that the curvature changes sign at the point. See

inpolar quadric: See

inscribed: Having vertices on a curve, as in inscribed figure. Tangent to some lines, as in inscribed circle.

integral: An integral is (more or less) what is now called a closed differential form, or sometimes the result of integrating such a form. An integral of the first kind is a holomorphic closed differential form. An integral of the second kind is a meromorphic closed differential form with no residues. An integral of the third kind is a meromorphic closed differential form whose poles are all simple. A simple integral is a closed 1-form, or the result of integrating a 1-form. A double integral is a closed 2-form, or the result of integrating a 2-form.

invariant: (Noun) A polynomial in the coefficients of a homogeneous form, invariant under some group of linear transformations. See also covariant, contravariant, concomitant.

inversion: An inversion is a transformation of order 2 exchanging the inside and outside of a circle. See.

involute: An involute is a curve obtained by unrolling a string around a curve. See.

involution: A transformation whose square is the identity. Cremona transformations that are involutions include Bertini involutions, Geiser involutions, and De Jonquières involutions.

irregularity: The irregularity of a surface is the dimension of the space of holomorphic 1-forms on a non-singular projective surface; see Hodge number.

isologue: Given a Cremoma transformation T, the isologue of a point p is the set of points x such that p, x, T(x) are collinear. The point p is called the center of the isologue.

J
Jacobian: The Jacobian variety of a curve A Jacobian curve; see below

Jacobian curve: The locus of double points of curves of a net.

Jacobian set: The set of free double points of a pencil of curves.

Jacobian system: The linear system generated by Jacobian curves.

join: The join of two linear spaces is the smallest linear space containing both of them.

K
kenotheme: An intersection of n hypersurfaces in n-dimensional projective space. Archaic.

keratoid: Horn-like. A keratoid cusp is one whose two branches curve in opposite direction; see ramphoid cusp.

Kirkman point: One of the 60 points lying on 3 of the Plücker lines associated with 6 points on a conic.

Klein: Felix Klein The Klein icosahedral surface is a certain cubic surface The Klein quartic is the curve $x^3y + y^3z + z^3x = 0.$

Kronecker index: The intersection number of two curves on a surface

Kummer surface: A quartic surface with 16 nodes

L
Laguerre net: A net V of plane curves of some degree d such that the base locus of a generic pencil of V is the base locus of V together with d–1 collinear points

lemniscate: A lemniscate is a curve resembling a figure 8. See

limaçon: A limaçon is a curve traced by a point on a circle rolling around a similar circle. See

line: A line in projective space; in other words a subvariety of degree 1 and dimension 1.

line coordinates: Projective coordinates. See

linear: Degree 1

linear system: A linear system of divisors, given by the zeros of elements of a vector space of sections of a line bundle

locus: 1-A subset of projective space given by points satisfying some condition

M
manifold: An algebraic manifold is a cycle of projective space, in other words a formal linear combination of irreducible subvarieties. Algebraic manifolds may have singularities, so their underlying topological spaces need not be manifolds in the sense of differential topology.

meet: The meet of two sets is their intersection.

Möbius tetrads: Two tetrads such that the plane containing any three points of one tetrad contains a point of the other.

model: A variety whose points (or sometimes hyperplane sections) correspond to elements of some family. Similar to what is now called a parameter space or moduli space. A model for a field extension K of a field k is a projective variety over k together with an isomorphism between K and its field of rational functions.

modulus: A function of algebraic varieties depending only on the isomorphism type; in other words, a function on a moduli space

Moebius tetrads: See

monoid: A surface of degree n with a point of multiplicity n–1.

monoidal transformation: A Cremona transformation of projective space generated by a family of monoids with the same point of multiplicity n–1. More generally a blow-up along a subvariety, called the center of the monoidal transformation.

multiple: A multiple point is a singular point (one with a non-regular local ring).

multiplicity: The multiplicity of a point on a hypersurface is the degree of the first non-vanishing coefficient of the Taylor series at the point. More generally one can define the multiplicity of any point of a variety as the multiplicity of its local ring. A point has multiplicity 1 if and only if it is non-singular.

N
Néron–Severi group: The Néron–Severi group is the group of divisors module numerical equivalence.

nest: Two components (circuits) of a real algebraic curve are said to nest if one is inside the other.

net: A 2-dimensional linear system. See "pencil" and "web". See also Laguerre net. A harmonic net is a set of points on a line containing the harmonic conjugate of any point with respect to any other two points.

Newton polygon: The convex hull of the points with coordinates given by the exponents of the terms of a polynomial.

nodal: A nodal tangent to a singular point of a curve is one of the lines of its tangent cone.

node: A singular point p of a hypersurface f = 0, usually with the determinant of the Hessian of f not zero at p.

node cusp: A singularity of a curve where a node and a cusp coincide at the same point.

normal:

A normal intersection is an intersection with the "expected" codimension (given a sum of codimensions).

null-polarity: A correlation given by a skew symmetric matrix. A null-polarity of the projective space of a vector space is essentially a non-degenerate skew-symmetric bilinear form, up to multiplication by scalars. See also polarity.

O
octad: A set of 8 points

octic: (Adjective) Degree 8 (Noun) A degree 8 projective variety

ombilic: The curve at infinity which is the intersection of any sphere with the plane at infinity. All points of the ombilic are non-real.

order: Now called degree of an algebraic variety: the number of intersection points with a generic linear subspace of complementary dimension. The order of a covariant or concomitant: its degree in the contravariant variables. The order of a Cremona transformation is the order (degree) of its homaloids.

ordinary: An ordinary point of multiplicity m of a curve is one with m distinct tangent lines.

oscnode: A double point of a plane curve that is also a point of osculation; in other words the two branches meet to order at least 3.

osculate: Kiss; to meet with high order. See.

osculating plane: A tangent plane of a space curve having third order contact with it.

outpolar quadric: See and

P
Pappus: Pappus of Alexandria. The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus's hexagon theorem.

parabolic point: A point of a variety that also lies in the Hessian.

parallel: Meeting at the line or plane at infinity, as in parallel lines A parallel curve is the envelope of a circle of fixed radius moving along another curve.

partitivity: The number of connected components of a real algebraic curve. See.

Pascal: Short for Pascal line, the line determined by 6 points of a conic in Pascal's theorem

pedal: The pedal curve of C with respect to a pedal point P is the locus of points X such that the line through X orthogonal to PX is tangent to C.

pencil: A 1-dimensional linear system. See pencil (mathematics) and Lefschetz pencil.

pentad: A set of 5 points

pentahedron: A union of 5 planes, in particular the Sylvester pentahedron of a cubic surface.

period: The integral of a differential form over a submanifold

perspectivity: An isomorphism between two projective lines (or ranges) of projective space such that the lines joining each point of one line to the corresponding point of the other line all pass through a fixed point, called the center of the perspectivity or the perspector.

perspector: The center of a perspectivity

perspectrix: The line in Desargues theorem on which the intersections of pairs of sides of two perspective triangles lie

pinch: A pinch point is a singular point of a surface, where the two tangent planes of a point on a double curve coincide in a double plane, called the pinch plane.

pippian: Introduced by. Now called the Cayleyan. See also quippian.

Plücker:

For Plücker characteristic see characteristic A Plücker line is one of the 15 lines containing 4 of the 20 Steiner points associated to 6 points on a conic. The Plücker lines meet in threes at the 60 Kirkman points.

plurigenus: Plural plurigenera The dth plurigenus of a variety is the dimension of the space of sections of the dth power of the canonical line bundle.

point-star: A family of lines with a common point

polar: (Adjective) Related by a polarity The polar conic is the zero set of the quadratic form associated to a polarity, or equivalently the set of self-conjugate points of the polarity. (Noun) The first polar, second polar, and so on are varieties of degrees n–1, n–2, ... formed from a point and a hypersurface of degree n by polarizing the equation of the hypersurface. A polar or polar line is the line corresponding to a point under a polarity of the projective plane.

polarity: A correlation given by a symmetrical matrix, or a correlation of period 2. A polarity of the projective space of a vector space is essentially a non-degenerate symmetric bilinear form, up to multiplication by scalars. See also null-polarity.

pole: The point corresponding to a hyperplane under a polarity. A singularity of a rational function.

poloconic: polocubic: poloquartic: The poloconic (also called conic polar) of a line in the plane with respect to a cubic curve is the locus of points whose first polar is tangent to the line.

polygonal: A polygonal (or k-gonal) curve is a curve together with a map (of degree k) to the projective line. The degree of the map is called the gonality of the curve. When the degree is 1, 2, or 3 the curve is called rational, hyperelliptic, or trigonal.

porism: A porism is a corollary, especially in geometry, as in Poncelet's porism. The precise meaning seems to be controversial. An arrangement of geometrical figures (such as lines or circles) that are inscribed in one curve and circumscribed around another, as in Poncelet's porism or Steiner's porism. There seems to be some confusion about whether "porism" refers to the geometrical configuration or to the statement of the result. poristic: Having either no solutions or infinitely many. For example, Poncelet's porism and Steiner's porism imply that if there is one way to arrange lines or circles then there are infinitely many ways.

postulated: A postulated object (point, line, and so on) is an object in some larger space. For example, a point at infinity of projective space is a postulated point of affine space.

postulation: The postulation of a variety for some family is the number of independent conditions needed to force an elements of the family to contain the variety.

power of a point: Laguerre defined the power of a point with respect to an algebraic curve of degree n to be the product of the distances from the point to the intersections with a circle through it, divided by the nth power of the diameter. He showed that this is independent of the choice of circle through the point.

prime: An old term for a hyperplane in a projective space.

primal: An old term for a projective hypersurface.

projectivity: An isomorphism between two projective lines (or ranges). A projectivity is a product of at most three perspectivities.

propinquity: A number depending on two branches at a point, defined by.

proximate: For proximate points see.

pure: All components are of the same dimension. Now called equidimensional.

Q
quadratic transformation: A Cremona transformation of degree 2. A standard quadratic transformation is one similar to the map taking each coordinate to its inverse. A monomial transformation with center a point, or in other words a blowup at a point.

quadric: Degree 2, especially a degree 2 projective variety. Not to be confused with quantic or quartic.

quadrisecant: A quadrisecant is a line meeting something in four points

quadro-cubic, quadro-quartic: A quadro-cubic or quadro-quartic transformation is a Cremona transformation such that the homaloids of the transformation have degree 2 and those of its inverse have degree 3 or 4.

quantic: A homogeneous polynomial in several variables, now usually called a form. Not to be confused with quartic or quadric.

quarto-quartic: A Quarto-quartic transformation is a Cremona transformation such that the homaloids of the transformation and its inverse all have degree 4.

quaternary: Depending on four variables, as in quaternary form.

quartic: Degree 4, especially a degree 4 projective variety. Not to be confused with quantic or quadric.

quintic: Degree 5, especially a degree 5 projective variety.

quippian: A quippian is a degree 5 class 3 contravariant of a plane cubic introduced by and discussed by. See also pippian.

quotient ring: The quotient ring of a point (or more generally a subvariety) is what is now called its local ring, formed by adding inverses to all functions that do not vanish identically on it.

R
ramphoid: Beak-like. A ramphoid cusp is one whose two branches curve in the same direction; see keratoid cusp.

rank: The rank of a projective curve is the number of tangents to the curve meeting a generic linear subspace of codimension 2. The rank of a projective surface is the rank of a curve given by the intersection of the surface with a generic hyperplane. See order, class, type.

range: The set of all points on a line. A labeled or finite ordered set of points on a line.

rational: Birational to projective space. Defined over the rational numbers.

ray: A line, especially one in a family of lines

regular:

Symmetrical, as in regular polygon, regular polyhedron. Defined everywhere, as in regular (birational) map.

regulus: One of the two pencils of lines on a product of two projective planes or a quadric surface.

related: Two ranges (labeled sets) of points on a line are called related if there is a projectivity taking one range to the other.

representative manifold: A parameter space or moduli space for some family of varieties

residual: The residual intersection of two varieties consists of the "non-obvious" part of their intersection.

resultant: The resultant of two polynomials, given by the determinant of the Sylvester matrix of two binary forms, that vanishes if they have a common root. A Cremona transformation formed from n correlations of n-dimensional projective space.

reverse: Inverse (of a function or birational map)

ruled: Covered by lines, as in ruled surface. See also scroll.

S
s: Projective space of dimension n.

Salmon conic: The Salmon conic of a pair of plane conics is the locus of points such that the pairs of tangents to the two conics are harmonically conjugate.

satellite: If a line meets a cubic curve in 3 points, the residual intersections of the tangents of these points with the cubic all lie on a line, called the satellite line of the original line. See. A certain plane curve of degree (n–1)(n–2) constructed from a plane curve of degree n and a generic point. For satellite points see. Possibly something to do with base points.

scroll: A ruled surface with an embedding into projective space so that the lines of the ruled surface are also lines of projective space.

secant: A line intersecting a variety in 2 points, or more generally an n-dimensional projective space meeting a variety in n+1 points. A secant variety is the union of the secants of a variety.

second kind: All residues at poles are zero

secundum: An intersection of two primes (hyperplanes) in projective space.

Segre: Named after either Beniamino Segre or Corrado Segre A Segre variety or Segre embedding is the product of two projective spaces, or an embedding of this into a larger projective space. The Segre cubic is a cubic hypersurface in 4-dimensional projective space.

self-conjugate: self-polar: Incident with its image under a polarity. In particular the self-conjugate points of a polarity form the polar conic. A self-conjugate (or self-polar) triangle (or triad) is a triangle such that each vertex corresponds to the opposite edge under a polarity. A self-conjugate tetrad is a set of 4 points such that the pole of each side lies on the opposite side.

septic: septimic: (Adjective) Degree 7 (Noun) A degree 7 projective variety (Noun) A degree 7 form

sextactic point: One of the 27 points of an elliptic curve of order dividing 6 but not 3.

sextic: Degree 6, especially a degree 6 projective variety

simple: A simple point of a variety is a non-singular point. More generally a simple subvariety W of a variety V is one with a regular local ring, which means roughly that most points of W are simple points of V.

singular: Special in some way, including but not limited to the current sense of having a singularity

skew: Intersecting in a set that is either empty or of the "expected" dimension. For example skew lines in projective 3-space do not intersect, while skew planes in projective 4-space intersect in a point.

solid: A 3-dimensional linear subspace of projective space, or in other words the 3-dimensional analogue of a point, line, or plane.

special divisor: An effective divisor whose first cohomology group (of the associated invertible sheaf) is non-zero.

spinode: A cusp. ,

star: A collection of lines (and sometimes planes and so on) with a common point, called the center of the star.

stationary point: A cusp. See.

Steiner: Steinerian: Named after Jakob Steiner A Steinerian is the locus of the singular points of the polar quadrics of a hypersurface. A Steiner surface is a certain embedding of the projective plane into projective 3-space. a Steiner point is one of the 20 points lying on 3 of the Pascal lines associated with 6 points on a conic.

Steiner–Hessian: One of Cayley's names for the Cayleyan. See.

surface: An abstract surface together with an embedding into projective space.

superabundance of a divisor on a surface.: The dimension of the first cohomology group of the corresponding sheaf.

symmetroid: The zeros of the determinant of a symmetric matrix of linear forms

syntheme: A partition of a set of 6 elements into 3 pairs, or an element of the symmetric group on 6 points of cycle shape 222.

system: A family of algebraic sets in projective space; for example, a line system is a family of lines.

syzygetic: Paired. Opposite of azygetic, meaning unpaired. Example: syzygetic triad, syzygetic tetrad, syzygetic set, syzygetic pencil.

syzygy: A point is in syzygy with some other points if it is in the linear subspace generated by them. A syzygy is a linear relation between points in an affine space. An algebraic relation between generators of a ring, especially a ring of invariants or covariants. A linear relation between generators of a module, or more generally an element of the kernel of a homomorphism of modules. A global syzygy is a resolution of a module or sheaf.

T
tacnode: A tacnode is a point of a curve where two branches meet in the same direction.

tacnode-cusp: A singularity of a plane curve where a tacnode and a cusp are combined at the same point.

tact-invariant: An invariant of two curves that vanishes if they touch each other. See.

tangent cone: A tangent cone is a cone defined by the non-zero terms of smallest degree in the Taylor series at a point of a hypersurface.

tangential equation: The tangential equation of a plane curve is an equation giving the condition for a line to be tangent to the curve. In other words it is the equation of the dual curve. It is not the equation of a tangent to a curve.

ternary: Depending on three variables, as in ternary form

tetrad: A set of 4 points

tetragram: Synonym for complete quadrilateral

tetrahedroid: A tetrahedroid is a special kind of Kummer surface.

tetrahedron: A geometric configuration consisting of 4 points and the 6 lines joining pairs. This is similar to the lines and infinite edges of a polyhedral tetrahedron, but in algebraic geometry one sometimes does not include the faces of the tetrahedron.

tetrastigm: Synonym for complete quadrangle

third kind: All poles are simple (order 1)

threefold: (Adjective) Three-dimensional (Noun) A 3-dimensional variety

torsal generator.: A generator of a scroll (ruled surface) that meets its consecutive generator. See.

torse: Developable surface. transvectant: An invariant depending on two forms.

transversal: A line meeting several other lines. For example, 4 generic lines in projective 3-space have 2 transversals meeting all of them.

triad: A set of 3 points

tricircular: A tricircular curve is one that passes through the circular points at infinity with order 3.

tricuspidal: Having three cusps

trigonal: A trigonal curve is one with a degree three map to the projective line. See hyperelliptic.

trihedral: A set of 3 planes A Steiner trihedral is a set of three tritangent planes of a cubic surface whose intersection point is not on the surface.

trilinear coordinates: Coordinates based on distance from sides of a triangle: Trilinear coordinates.

trinodal: Having three nodes

tripartite: Having three connected components.

trisecant: A line meeting a variety in 3 points. See trisecant identity.

tritangent: Meeting something in 3 tangent points, such as a tritangent conic to a cubic curve or a tritangent plane of a cubic surface.

trope: A trope is a singular (meaning special) tangent space. The word is mostly used for a tangent space of a Kummer surface touching it along a conic.

twisted: A twisted cubic is a degree 3 embedding of the projective line in projective 3-space

total: A set of 5 partitions of a 6-element set into three pairs, such that no two elements of the total have a pair in common. For example, {(12)(36)(45), (13)(24)(56), (14)(26)(35), (15)(23)(46), (16)(25)(34)}

type: The type of a projective surface is the number of tangent planes meeting a generic linear subspace of codimension 4.

U
undulation: A point of undulation of a curve is where the tangent meets the curve to fourth order; also called a hyperflex. See inflection point.

unibranch: Having only one branch at a point. For example, a cusp of a plane curve is unibranch, while a node is not.

unicursal: A unicursal curve is one that is rational, in other words birational to the projective line. See.

unipartite: Connected. See

unirational: A correspondence is called unirational if it is generically injective, in other words a rational map. A variety is called unirational if it is finitely covered by a rational variety.

united point: A point in the intersection of the diagonal and a correspondence from a set to itself.

unode: A double point of a surface whose tangent cone consists of one double plane. See binode.

V
valence: valency: The valence or valency of a correspondence T on a curve is a number k such that the divisors T(P)+kP are all linearly equivalent. A correspondence need not have a valency.

Veronese surface: An embedding of the projective plane in 5-dimensional projective space.

virtual: An estimate for something that is often but not always correct, such as virtual genus, virtual dimension, and so on. If some number is given by the dimension of a space of sections of some sheaf, the corresponding virtual number is sometimes given by the corresponding Euler characteristic, and equal to the dimension when all higher cohomology groups vanish. See superabundance.

W
web: A 3-dimensional linear system. See "net" and "pencil".

Weddle surface: A quartic surface in projective space given by the locus of the vertex of a cone passing through 6 points in general position.

Weierstrass point: A point on a curve where the dimension of the space of rational functions whose only singularity is a pole of some order at the point is higher than normal.

Wirtinger sextic: A degree 4 genus 6 plane curve with nodes at the 6 points of a complete quadrangle.

XYZ
Zeuthen–Segre invariant: The Zeuthen–Segre invariant is 4 less than the Euler characteristic of a non-singular projective surface.