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. Dolgachev (2012) translates many of the classical terms in algebraic geometry into scheme-theoretic terminology. Other books defining some of the classical terminology include Baker (1922a, 1922b, 1923, 1925, 1933a, 1933b), Coolidge (1931), Coxeter (1969), Hudson (1990), Salmon (1879), Semple & Roth (1949).
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. (Semple & Roth 1949, p.20–21) 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 , … , n {\displaystyle 1,2,\ldots ,n} . This notation was introduced by Schubert (1886). ∞¹, ∞², ... A family of dimension 1, 2, ... {1}, {2}, ...,{n} A family or variety of dimension 1 , 2 , … , n {\displaystyle 1,2,\ldots ,n} . (Semple & Roth 1949, p.288)
A
Abelian group 1. An archaic name for the symplectic group. 2. A commutative group.
aberrancy The deviation of a curve from circular form. See Salmon (1879, p. 356).
absolute 1. 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. 2. 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. (Baker 1933b, vol 6, p. 157)
acnode An acnode is an isolated point of a real curve. See Salmon (1879, p.23).
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. (Semple & Roth 1949, p.55, 231)
affine 1. Affine space is roughly a vector space where one has forgotten which point is the origin. 2. 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 Salmon (1879, p.119).
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. (Semple & Roth 1949, p.55, 231)
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 1. 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. (Semple & Roth 1949, p.274) Similar to directrix.
azygetic Unpaired. Opposite of syzygetic, meaning paired. Example: azygetic triad, azygetic tetrad, azygetic set.
B
base 1. A base point is a point common to all members of a family. 2. 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 Salmon (1879, p.231).
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 1. A bielliptic curve is a branched double cover of an elliptic curve. 2. A bielliptic surface is the same as a hyperelliptic surface.
bifid 1. Split into two equal parts 2. 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}. (Dolgachev 2012, p.215) 3. 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 Salmon (1879, p.223).
biflecnode Same as fleflecnode. See Salmon (1879, p.210).
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. (Semple & Roth 1949, p.424)
bipartite Having two connected components. See Salmon (1879, p.165).
bipunctual 1. Having two points 2. For a bipunctual conic with respect to 3 points see Baker (1922b, vol 2, p. 123).
birational 1. Two varieties are birational if they are isomorphic off lower-dimensional subsets 2. A birational map is a rational map with rational "inverse"
biregular 1. A biregular map is a regular map with regular inverse 2. 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. (Dolgachev 2012)
bitangent A bitangent is a line that is tangent to a curve at two points. See Salmon (1879, p. 328).
bitangential Meeting a curve at the tangency points of its bitangents
Brianchon hexagon A non-planar hexagon whose three diagonals meet. (Baker 1922a, vol 1, p. 47)
C
canonical 1. The canonical series is the linear series of the canonical line bundle 2. The canonical bundle is the line bundle of differential forms of highest degree. 3. The canonical map or canonical embedding is the map to the projective space of the sections of the canonical bundle 4. A canonical curve (or variety) is the image of a curve (or variety) under the canonical map 5. The canonical class is the divisor class of a canonical divisor 6. A canonical divisor is a divisor of a section of the canonical line bundle.
Canonizant A canonizant is a covariant of forms.
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 1. See Salmon (1879) 2. A Cayley octad is a set of 8 points in projective space given by the intersection of three quadrics. (Dolgachev 2012, 6.3.1) 3. The Cayley lines or Cayley–Salmon lines are the 20 lines passing through 3 Kirkman points. 4. A Cayley absolute is a conic or quadric used to define a metric.
center centre 1. A special point associated with some geometric object 2. The center of a perspectivity 3. The center of an isologue
character characteristic 1. An integer associated with a projective variety, such as its degree, rank, order, class, type. (Semple & Roth 1949, p.189) 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. (Coolidge 1931, p.99) 2. 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. (Coolidge 1931, p.220) 3. 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. (Coolidge 1931, p. 50)
circular 1. A circular point is one of the two points at infinity (1: i: 0), (1: −i: 0) through which all circles pass 2. A circular algebraic curve is a curve passing through the two circular points at infinity. See also bicircular.
circumscribed 1. Having edges tangent to some curve, as in circumscribed quadrilateral. 2. Passing through the vertices of something, as in circumscribed circle.
cissoid A cissoid is the curve generated from two curves and a point. See Salmon (1879).
class 1. The class of a plane curve is the number of proper tangents passing through a generic point of the plane. (Semple & Roth 1949, p.28) 2. The class of a space curve is the number of osculating planes passing through a generic point of space. (Semple & Roth 1949, p.85) 3. The class of a surface in rdimensional projective space is the number of tangent planes meeting a generic codimension 2 subspace in a line. (Semple & Roth 1949, p.28) 4. 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). (Baker 1922b, vol 2, p. 66)
coincidence 1. A coincidence quadric is a quadric associated to a correlation, given by the locus of points lying in the corresponding hyperplane. (Semple & Roth 1949, p.8) 2. A fixed point of a correspondence, in other words a point of a variety corresponding to itself under a correspondence. (Coolidge 1931, p. 126)
collinear On the same line
collineation A collineation is an isomorphism from one projective space to another, often to itself. (Semple & Roth 1949, p.6) See correlation.
complete 1. A linear series of divisors is called complete if it is not contained in a larger linear series.(Semple & Roth 1949, p.351) 2. A scheme is called complete if the map to a point is proper 3. A complete quadrangle is 4 points and the 6 lines joining pairs 4. A complete quadrilateral is 4 lines meeting in pairs in 6 points 5. 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 1. (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. (Semple & Roth 1949, p.236) See congruence. 2. (Adjective.) Related to the complex numbers. 3. 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 Salmon (1879).
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⊕V⊕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 1. 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, ...(Semple & Roth 1949, p.18) 2. 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 (Semple & Roth 1949, p.238, 288). See complex.
conic A conic is a degree 2 curve. Short for "conic section", the intersection of a cone with a plane.
conjugate 1. A conjugate point is an acnode. (Salmon 1879, p.23) 2. A conjugate point is a point lying on the hyperplane corresponding to another point under a polarity. 3. A conjugate line is a line containing the point corresponding to another line under a polarity (or plane conic). (Baker 1922b, vol 2, p. 26) 4. 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 1. 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⊕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. (Semple & Roth 1949, p.7)
coresidual See Salmon (1879, p.131)
correspondence A correspondence from X to Y is an algebraic subset of X×Y
cosingular Having the same singularities
couple An ordered pair
covariant 1. 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⊕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 2. The variety defined by a covariant. In particular the curve defined by the Hessian or Steinerian covariants of a curve are called covariant curves. (Coolidge 1931, p.151)
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. Semple & Roth (1949, p.179)
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 (Semple & Roth 1949, p.85, 87)
cyclide A cyclide is a quartic surface passing doubly through the absolute conic. (Semple & Roth 1949, p.141)
D
decic decimic 1. (Adjective) Degree 10 2. (Noun) A degree 10 projective variety
deficiency 1. The deficiency of a linear system is its codimension in the corresponding complete linear system. 2. 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. (Semple & Roth 1949, p.30), (Salmon 1879, p. 28)
degree 1. The number of intersection points of a projective variety with a generic linear subspace of complementary dimension 2. 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 1. (Noun) A 1-dimensional family of planes in 3-dimensional projective space (Semple & Roth 1949, p.85). 2. (Noun) The envelope of the normals of a curve 3. (Noun) Short for a developable surface, one that can be unrolled to a plane 4. The tangent developable of a curve is the surface consisting of its tangent lines. 5. Flat, as in developable surface
differential 1. A differential of the first kind is a holomorphic 1-form. 2. A differential of the second kind is a meromorphic 1-form such that the residues of all poles are 0. Sometimes it is only allowed to have one pole that must be of order 2. 3. 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. (Baker 1922b, vol 2, p. 26)
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 1. A 0-dimensional singularity of multiplicity 2, such as a node. One of the two points fixed by an involution of a projective line. (Baker 1922b, vol 2, p.3)
double six The Schläfli double six configuration
duad A set of two points
dual 1. The dual of a projective space is the set of hyperplanes, considered as another projective space. 2. The dual curve of a plane curve is the set of its tangent lines, considered as a curve in the dual projective plane. 3. A dual number is a number of the form a+εb where ε has square 0. Semple & Roth (1949, p.268)
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. (Baker 1922b, vol 2, p. 49)
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 Salmon (1879, p. 65).
epitrochoid An epitrochoid is the curve traced by a point of a disc rolling along another disc. Salmon (1879)
equiaffine equiaffinity An equiaffinity is an equiaffine transformation, meaning an affine transformation preserving area.
equianharmonic 1. Four points whose cross ratio (or anharmonic ratio) is a cube root of 1 2. 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 Salmon (1879, p. 184).
evolute An evolute is the envelope of the normal lines of a plane curve. See Salmon (1879, p. 40).
exceptional 1. Corresponding to something of lower dimension under a birational correspondence, as in exceptional curve, exceptional divisor 2. 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. (Salmon 1885, p.243)
first kind holomorphic or regular (when applied to differentials)
flat 1. (Noun) A linear subspace of projective space, such as a point, line, plane, hyperplane. 2. (Adjective) Having curvature zero. 3. (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. (Cayley 1852). (Salmon 1879, p.210)
fleflecnode A double point that is also a point of inflexion of both branches. (Cayley 1852).
flex Short for point of inflection
focal 1. A focal point, line, plane, ... is the intersection of several consecutive elements of a family of linear subspaces. (Semple & Roth 1949, p. 85, 252) 2. A focal curve, surface and so on is the locus of the focal points of a family of linear subspaces. (Semple & Roth 1949, p.252)
focus A focal point. See Salmon (1879, p. 116), (Semple & Roth 1949, p. 85,251)
foliate singularity See (Semple & Roth 1949, p.422)
form 1. A homogeneous polynomial in several variables. Same as quantic. 2. 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. (Semple & Roth 1949, p.26).
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". 1. 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. 2. A fundamental point, curve, or variety is a point, curve, or variety in the fundamental set of a birational correspondence.
G
grd, γrd 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 (Semple & Roth 1949, p.204) or more generally an element of some family of linear spaces.
generic 1. Not having some special properties, which are usually not stated explicitly. 2. A generic point is one having coordinates that are algebraically independent over the base field. 3. The generic point of a scheme.
genus 1. The dimension of the space of sections of the canonical bundle, as in the genus of a curve or the geometric genus of a surface 2. arithmetic genus of a surface 3. plurigenus
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 (Semple & Roth 1949, p.345), and the grade of a net of curves on a surface is the number of free intersections of two generic curves. (Semple & Roth 1949, p.45) (Semple & Roth 1949, p.159)
Grassmannian A Grassmannian is a variety parameterizing linear subspaces of projective space
group 1. 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. (Coolidge 1931) 2. A group in the usual sense.
H
harmonic 1. 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. 2. 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. 3. Satisfying some analogue of the Laplace equation, as in harmonic form. 4. The harmonic polar line of an inflection point of a cubic curve is the component of the polar conic other than the tangent line. (Dolgachev 2012, 3.1.2) 5. 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. (Baker 1922a, vol 1, p. 133) 6. For harmonically conjugate conics see (Baker 1922b, vol 2, p. 122).
Hesse Hessian Named after Otto Hesse. 1. A Hessian matrix, or a variety associated with it. See Salmon (1879, p.55). 2. 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. 3. The Hessian point is a point associated to three lines tangent to a conic, whose construction is dual to that of a Hessian line. 4. 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. 5. The Hesse configuration is the configuration of inflection points of a plane cubic. 6. 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 1. 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. (Semple & Roth 1949, p.45) (Coolidge 1931, p. 442) When the linear system has dimension 2 or 3 it is called a homaloidal net or homaloidal web. 2. Homaloidal means similar to a flat plane.
homographic 1. Having the same invariants. See Salmon (1879, p.232). 2. A homographic transformation is an automorphism of projective space over a field, in other words an element of the projective general linear group. (Salmon 1879, p.283)
homography 1. An isomorphism between projective spaces induced by an isomorphism of vector spaces. 2. An axis of homography is a line associated to two related ranges of a conic. (Baker 1922b, vol 2, p. 16)
homology 1. As in homology group 2. 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. 3. 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 Salmon (1879, p.175).
hypercusp A singularity of a curve of some multiplicity r whose tangent cone is a single line meeting the curve with order r+1. (Coolidge 1931, p. 18)
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). Semple & Roth (1949, p.381)
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 Salmon (1879, p. 32)
inpolar quadric See (Baker 1923, vol 3, p. 52, 88)
inscribed 1. Having vertices on a curve, as in inscribed figure. 2. 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.. 1. An integral of the first kind is a holomorphic closed differential form. 2. An integral of the second kind is a meromorphic closed differential form with no residues. 3. An integral of the third kind is a meromorphic closed differential form whose poles are all simple. 4. A simple integral is a closed 1-form, or the result of integrating a 1-form. 5. 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 Salmon (1879, p.103).
involute An involute is a curve obtained by unrolling a string around a curve. See Salmon (1879, p. 278).
involution 1. 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 1. The Jacobian variety of a curve 2. A Jacobian curve; see below
Jacobian curve The locus of double points of curves of a net. (Semple & Roth 1949, p.115)
Jacobian set The set of free double points of a pencil of curves. (Semple & Roth 1949, p.119)
Jacobian system The linear system generated by Jacobian curves. (Semple & Roth 1949, p.117)
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. (Sylvester 1853, Glossary p. 543–548) Archaic.
keratoid Horn-like. A keratoid cusp is one whose two branches curve in opposite direction; see ramphoid cusp. Salmon (1879)
Kirkman point One of the 60 points lying on 3 of the Plücker lines associated with 6 points on a conic.
Klein 1. Felix Klein 2. The Klein icosahedral surface is a certain cubic surface 3. The Klein quartic is the curve x 3 y + y 3 z + z 3 x = 0. {\displaystyle x^{3}y+y^{3}z+z^{3}x=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 (Dolgachev 2012, theorem 7.3.5) (Coolidge 1931, p. 423)
lemniscate A lemniscate is a curve resembling a figure 8. See Salmon (1879, p.42)
limaçon A limaçon is a curve traced by a point on a circle rolling around a similar circle. See Salmon (1879, p.43)
line A line in projective space; in other words a subvariety of degree 1 and dimension 1.
line coordinates Projective coordinates. See Salmon (1879, p. 7)
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. Semple & Roth (1949, p.14–15)
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. (Baker 1922a, vol 1, p. 62) model 1. 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. 2. 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 Möbius tetrads
monoid A surface of degree n with a point of multiplicity n–1. (Semple & Roth 1949, p.187)
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. (Semple & Roth 1949, p.187)
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. (Coolidge 1931)
net 1. A 2-dimensional linear system. See "pencil" and "web". See also Laguerre net. 2. 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. (Baker 1922a, vol 1, p. 133)
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. (Semple & Roth 1949, p.26)
node A singular point p of a hypersurface f = 0, usually with the determinant of the Hessian of f not zero at p. (Cayley 1852)
node cusp A singularity of a curve where a node and a cusp coincide at the same point. (Salmon 1879, p. 207)
normal 1. A subvariety of projective space is linearly normal if the linear system defining the embedding is complete; see rational normal curve. 2. Orthogonal to the tangent space, such as a line orthogonal to the tangent space or the normal bundle. 3. A normal intersection is an intersection with the "expected" codimension (given a sum of codimensions). (Semple & Roth 1949, p.16) 4. Local rings are integrally closed; s