Quadratic set

In synthetic geometry , the term quadratic set describes sets, which are referred to as projective quadrics in analytical geometry , without coordinates , solely through their incidence and richness properties. He generalizes this term in such a way that it can also be used for non-Desarguean projective planes and for non- Pappus projective geometries . Quadratic sets and their tangent spaces are themselves geometries in a more general sense, so-called incidence structures , in some cases they are even projective geometries. The term is particularly useful for finite geometries.

Elliptical circle after Frans van Schooten . The hinge mechanism was invented by the Dutch mathematician in the 17th century. If you drag the pen at point E, it will draw an ellipse. The mechanism is attached to the drawing pad at focal points H and I. The construction is based on the definition of the ellipse as a locus.

history

Quadrics in the plane of the drawing, especially ellipses , have been explored at least since classical antiquity . Until the 18th century, they were defined by describing their construction with the help of drawing devices (see the figure at the end of the introduction) or as a geometric location largely without reference to a coordinate system. One could therefore speak of a “synthetic” concept of quadrics for this period. However, it wasn't until the 19th century that an axiomatic basis for projective geometry was developed. Previously, as a geometry descriptive, it had only consisted of language rules for “improper” objects that were “added” to the plane of drawing or the visual space. Since the turn of the 20th century, non-desargue projective planes have been known, until the 1960s a large number of (especially finite) models for such planes were found. The analytical description of quadrics as a set of zeros in quadratic coordinate equations , which has led to a satisfactory algebraic classification of all quadrics for Pappusian geometries (see principal axis transformation , projective quadric ), can only be used to a limited extent for geometries over non-commutative oblique bodies ; it is largely applicable to non-desargus planes useless. The term "square set" was introduced by Buekenhout in 1969 in order to be able to describe quadrics in such planes. Quadrics have been systematically examined in this way since the 1970s. Since the finite projective planes also play an important role in coding theory , results with surprising applications seem to be found in this context from time to time, apparently far removed from abstract geometry.

Definitions

An (unlimited) double cone in the Euclidean intuition space is a quadratic set in its projective conclusion and is one of the cones . This quadratic amount has degenerated , because the tangential space of the tip of the cone is the total space, more precisely the radical of the cone consists precisely of this tip. If the cone is intersected with a plane that does not contain the tip of the cone, then a non-degenerate quadratic set is created in this plane. The three cases shown, parabola (A), ellipse (B) and hyperbola (C) differ only insignificantly as square sets in the projective closure of the respective cutting plane: each of these square sets is an oval . One of the degenerate quadratic sets of points, straight lines or double straight lines is induced in the same way in cutting planes containing the apex of the cone.

Quadratic set, tangent

Let be a projective geometry of arbitrary, finite dimension and let be a set of points of this geometry. ${\displaystyle \mathbb {P} }$ ${\displaystyle {\mathcal {Q}}}$

1. If a line of the geometry either has in common only a point or if any point of in is included, it means a tangent to . ${\displaystyle g}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle g}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle g}$ ${\displaystyle {\mathcal {Q}}}$
2. A tangent to that has only one point in common is called a tangent to in . ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle P}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle P}$
3. A tangent with the property that each point of in is called -Straight, more generally called a subspace one - subspace , if every point of in is included. ${\displaystyle g}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle U}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle U}$ ${\displaystyle {\mathcal {Q}}}$
4. For each point , the set consisting of the point and all points connected by a tangent is called the tangent space from on . This tangent space is also noted as. ${\displaystyle P\in {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}_{P}}$ ${\displaystyle P}$ ${\displaystyle A\neq P}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathfrak {T}}({\mathcal {Q}},P)}$

The set is called the quadratic set of if both of the following conditions are met: ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle \mathbb {P} }$

1. (“If 3 then all!”) Every straight line that contains at least three points of is entirely contained in. In other words: every straight line has no, exactly one, exactly two or all points in common. ${\displaystyle g}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$
2. (Tangent axiom) For every point the tangent space is the set of points of a hyperplane or the set of all points of . ${\displaystyle P\in {\mathcal {Q}}}$ ${\displaystyle {\mathfrak {T}}({\mathcal {Q}},P)}$ ${\displaystyle \mathbb {P} }$

Radical, degenerate square set

• For a quadratic set is the set of all points for which consists of all points of . This amount is called the radical of . ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle \mathop {\mathrm {Rad} } ({\mathcal {Q}})}$ ${\displaystyle P\in {\mathcal {Q}}}$ ${\displaystyle {\mathfrak {T}}({\mathcal {Q}},P)}$ ${\displaystyle \mathbb {P} }$ ${\displaystyle {\mathcal {Q}}}$
• A quadratic set is called non- degenerate if is, otherwise it is called degenerate . ${\displaystyle \mathop {\mathrm {Rad} } ({\mathcal {Q}})=\emptyset }$

Index of a quadratic set

• Let it be a quadratic set, the largest dimension of a subspace. Then the index is called by . The subspaces of the dimension are then also called maximum subspaces. ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle t-1}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle t}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle t-1}$ ${\displaystyle {\mathcal {Q}}}$

Oval and ovoid

→ Main article : Oval (projective geometry)

→ Main article : Ovoid (projective geometry)

• A non-empty set of points in a projective plane is called an oval if no three points are collinear and exactly one tangent goes through each point . ${\displaystyle {\mathcal {O}}}$ ${\displaystyle \mathbb {P} }$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {O}}}$

The generalization of the oval for spaces of any dimension is the ovoid :

• A non-empty set of points in a -dimensional space is called an ovoid if: ${\displaystyle {\mathcal {O}}}$ ${\displaystyle d}$ ${\displaystyle \mathbb {P} }$

1. No three points of are collinear, ${\displaystyle {\mathcal {O}}}$
2. for each point there is a hyperplane. ${\displaystyle P\in {\mathcal {O}}}$ ${\displaystyle {\mathfrak {T}}({\mathcal {Q}},P)}$

Nucleus and hyperoval

• If it exists, the common point of intersection of all tangents to an oval in a finite plane is called the nucleus of the oval.
• The set of points that make up an oval along with its nucleus. is called a hyperoval .

cone

Let be a hyperplane of projective space , a point that does not lie in and a non-degenerate, non-empty quadratic set of . Then it is called the quadratic set ${\displaystyle H}$ ${\displaystyle \mathbb {P} }$ ${\displaystyle S}$ ${\displaystyle H}$ ${\displaystyle {\mathcal {Q}}^{*}}$ ${\displaystyle H}$

${\displaystyle {\mathcal {Q}}:=\bigcup _{X\in {\mathcal {Q}}^{*}}(SX)}$

a cone with a point across . ${\displaystyle S}$ ${\displaystyle {\mathcal {Q}}^{*}}$

Elliptical, parabolic, and hyperbolic quadratic sets

Let be a non-degenerate quadratic set in a -dimensional projective geometry . Then the following terms are agreed: ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle d}$ ${\displaystyle \mathbb {P} }$

Main types of quadratic sets classified according to their index

Room dimension d	Index t	Name of the square set
${\displaystyle d}$ is just	${\displaystyle t={\frac {d}{2}}}$	parabolic
${\displaystyle d}$ is odd	${\displaystyle t={\frac {d-1}{2}}}$	elliptical
	${\displaystyle t={\frac {d+1}{2}}}$	hyperbolic

properties

index

Let be a quadratic set of the index in a -dimensional projective geometry . ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle t}$ ${\displaystyle d}$ ${\displaystyle \mathbb {P} }$

• Then a maximum subspace goes through each point of . ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$

• More precisely: Through every point from outside a -dimensional subspace there is a -dimensional -subspace , which intersects in a -dimensional subspace. ${\displaystyle P}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle t-1}$ ${\displaystyle U}$ ${\displaystyle t-1}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle V}$ ${\displaystyle t-2}$

If the quadratic set is non-degenerate and non-empty, then

• is if is even and ${\displaystyle t\leq {\frac {d}{2}}}$ ${\displaystyle d}$
• ${\displaystyle t\leq {\frac {d+1}{2}}}$if is odd. ${\displaystyle d}$

In addition, it is finite, then ${\displaystyle \mathbb {P} }$

• is if is even and ${\displaystyle t={\frac {d}{2}}}$ ${\displaystyle d}$
• ${\displaystyle t\in \lbrace {\frac {d-1}{2}},{\frac {d+1}{2}}\rbrace }$if is odd. ${\displaystyle d}$

In other words, in a finite projective geometry every non-degenerate and non-empty quadratic set

• parabolic if the dimension is even, ${\displaystyle d}$
• elliptical or hyperbolic if is odd. ${\displaystyle d}$

Classification of quadratic sets in the plane

Let be a quadratic set in a projective plane . Then the empty set, a one-point set, the point set of one or two straight lines, the entire point set or an oval. If and only if the quadratic set is non-empty and non-degenerate, it is an oval. ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle \mathbb {P} }$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$

Segre's theorem, quadratic sets and quadrics in pappus spaces

Let it be the d- dimensional, Pappusian projective space over a body K , whose characteristic is not 2. Then: ${\displaystyle \mathbb {P} =\mathbb {P} (K^{d+1})}$

• Each projective quadric of is a quadratic set. A quadric is non-degenerate as a square set if and only if the associated square form is non-degenerate. ${\displaystyle \mathbb {P} }$
• If and is a finite field , then every quadratic set is a projective quadric. ${\displaystyle d=2}$ ${\displaystyle K}$

The second statement follows from Segre's theorem :

Every oval in a finite Desarguessian plane of odd order is a conic section (in the sense of analytic geometry).

• In the finite planes of even order, there generally exist ovals that are not projective quadrics. The following applies more precisely: ${\displaystyle P(2,2^{r})}$
  1. In the Desargue's finite planes and , each oval is a projective quadric. ${\displaystyle P(2,2)}$ ${\displaystyle P(2,4)}$
  2. In every Desarguean finite plane of even order with there are ovals that are not projective quadrics. Each such oval is created from an oval , which is a projective quadric, by replacing any point on the quadric with the nucleus of that oval . ${\displaystyle P(2,2^{r})}$ ${\displaystyle r\geq 3}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$

Parabolic square set

Let be a parabolic square set in a 2- dimensional projective space . Then: ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle t}$ ${\displaystyle (t\geq 2)}$

1. If is a tangential hyperplane of , then the quadric induced in is a cone over a parabolic quadratic set. ${\displaystyle H}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle H}$ ${\displaystyle {\mathcal {Q}}^{*}={\mathcal {Q}}\cap H}$
2. If there is a hyperplane that is not a tangential hyperplane of , then the quadric induced in is an elliptical or hyperbolic quadric set. ${\displaystyle H}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle H}$ ${\displaystyle {\mathcal {Q}}^{*}={\mathcal {Q}}\cap H}$

Hyperbolic quadratic set

Let be a hyperbolic quadratic set in a -dimensional projective space . ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle 2t+1}$ ${\displaystyle (t\geq 2)}$

1. If is a tangential hyperplane of , then the quadric induced in is a cone over a hyperbolic quadratic set. ${\displaystyle H}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle H}$ ${\displaystyle {\mathcal {Q}}^{*}={\mathcal {Q}}\cap H}$
2. If there is a hyperplane that is not a tangential hyperplane of , then the quadric induced in is a parabolic quadratic set. ${\displaystyle H}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle H}$ ${\displaystyle {\mathcal {Q}}^{*}={\mathcal {Q}}\cap H}$

If a hyperbolic quadratic set exists in an at least three-dimensional projective space, then the space is Papposian, i.e. coordinated over a commutative body.

Numbers in finite spaces

Let it be a quadratic set in a -dimensional projective geometry over the finite body . For any point, let the number of lines pass through . Then: ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle d}$ ${\displaystyle \mathbb {P} =\mathbb {P} ({\mathbb {F} _{q}}^{d+1})}$ ${\displaystyle \mathbb {F} _{q}}$ ${\displaystyle P\in {\mathcal {Q}}\setminus \mathop {\mathrm {Rad} } ({\mathcal {Q}})}$ ${\displaystyle a_{P}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle P}$

1. The number is independent of the choice of ${\displaystyle a=a_{P}}$ ${\displaystyle P\in {\mathcal {Q}}\setminus \mathop {\mathrm {Rad} } ({\mathcal {Q}}).}$
2. If a hyperplane is, which is always the case, then contains exactly points of ${\displaystyle H={\mathfrak {T}}({\mathcal {Q}},P)}$ ${\displaystyle P\in {\mathcal {Q}}\setminus \mathop {\mathrm {Rad} } ({\mathcal {Q}})}$ ${\displaystyle H}$ ${\displaystyle aq+1}$ ${\displaystyle {\mathcal {Q}}.}$
3. The quadratic set contains exactly points. ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle 1+q^{d-1}+aq}$

Examples

An egg round in the real plane is an oval and a square set in the sense of the synthetic definition, but in general not a quadric!

• The empty set is a non-degenerate quadratic set in any projective geometry. In an at least one-dimensional projective space above the complex numbers , it is not a projective quadric. ${\displaystyle \mathbb {P} (\mathbb {C} ^{d+1}),d\geq 1}$
• Ovoids and ovoids in the conventional sense in real affine spaces, such as the “oval” in the figure on the right, are always quadratic sets in the projective closure of the space.

index

In two- or three-dimensional spaces, the following non-degenerate, non-empty, quadratic sets , which are quadrics, occur: ${\displaystyle {\mathcal {Q}}}$

• In two-dimensional spaces it always has the index 1 and is a parabolic oval , that is, the maximum dimension of contained subspaces is 0, individual points are the largest contained subspaces. In the affine classification there are 3 types: ellipse , parabola and hyperbola , but these are equivalent in the projective closure. ${\displaystyle {\mathcal {Q}}}$
• In three-dimensional spaces it has the index 1 or 2. ${\displaystyle {\mathcal {Q}}}$
  • Index 1 is elliptical. In the affine classification, it is an ellipsoid , a paraboloid or a two-shell hyperboloid , which are each projectively equivalent. ${\displaystyle {\mathcal {Q}}}$
  • Index 2 is hyperbolic: In the affine classification, it is a single-shell hyperboloid. Exactly two straight lines go through each point of - and this also applies to the projective closure . The totality of all straight lines is divided into two families, each of which creates the surface as a ruled surface . ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {Q}}}$

Solution numbers for homogeneous quadratic equations

• The equation describes a projective quadric, i.e. a square set , in every projective plane over a body . This has never degenerated - unless the characteristic is not 2. ${\displaystyle Q:A_{1}x_{1}^{2}+A_{2}x_{2}^{2}+A_{3}x_{3}^{2}=0;A_{1},A_{2},A_{3}\in K^{*}}$ ${\displaystyle K}$ ${\displaystyle {\mathcal {Q}}=\{\langle (x_{1},x_{2},x_{3})\rangle \in \mathbb {P} (K^{3})|A_{1}x_{1}^{2}+A_{2}x_{2}^{2}+A_{3}x_{3}^{2}=0\}}$ ${\displaystyle K}$

If the finite field with q elements ( q odd), then: ${\displaystyle K=\mathbb {F} _{q}}$

• The equation has a nontrivial solution, the quadratic set has the index 1 and is therefore an oval. ${\displaystyle Q}$ ${\displaystyle {\mathcal {Q}}}$
• ${\displaystyle {\mathcal {Q}}}$contains exactly projective points, three different points in are never collinear . ${\displaystyle q+1}$ ${\displaystyle {\mathcal {Q}}}$
• The equation has exactly nontrivial solutions. ${\displaystyle Q}$ ${\displaystyle (q-1)(q+1)=q^{2}-1}$

Compare with the existence statements formulated here Correlation (projective geometry) # polarities over finite spaces .

Fano level

Coordinated model of the Fano plane. The corners of the outer, isosceles triangle form a square set: an oval. Together with the point , these points form a four-point hyper-oval. ${\displaystyle 001,010,100}$ ${\displaystyle 111}$

In the Fano plane , the projective plane above the body with 2 elements , the set of zeros in the quadric is equal to the set of zeros in the straight line equation . The associated quadratic set is therefore a straight line and, like the quadrics and , which also describe straight lines, degenerated. ${\displaystyle K=\mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0}$ ${\displaystyle x_{1}+x_{2}+x_{3}=0}$ ${\displaystyle x_{1}^{2}+x_{2}^{2}=0}$ ${\displaystyle x_{1}^{2}=0}$

On the other hand, there is a quadric which is not equivalent to those mentioned. Its fulfillment set consists exactly of the projective points for which exactly one coordinate is not equal to 0, compare the figure, the quadratic set is an oval. The center of the triangle in the model is the intersection of all three tangents, so the corners together with the center form a hyper-oval. All ovals and hyper-ovals in the Fano plane emerge from this oval or hyper-oval through a projectivity . Hyperovals are exactly the complements of the seven straight lines, they are all complete quadrilaterals of the Fano plane. If you leave out an arbitrary point from such a hyperoval, you get a new oval equivalent to the one shown. ${\displaystyle x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{1}=0}$

literature

• Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry . From the basics to the applications (=  Vieweg Studium: advanced course in mathematics ). 2nd, revised and expanded edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X ( table of contents [accessed April 1, 2012]).
• Francis Buekenhout: Handbook of Incidence Geometry . North Holland, 1995, ISBN 0-444-88355-X .
• Burkhard Polster: A geometrical picture book . 1st edition. Springer, New York / Berlin / Heidelberg 1998, ISBN 0-387-98437-2 .

history

• Walter Benz: A Century of Mathematics, 1890–1990 . Festschrift for the anniversary of the DMV . Vieweg, Braunschweig 1990, ISBN 3-528-06326-2 .
• Gino Fano : Contrast of synthetic and analytical geometry in its historical development in the XIX. Century. In: Encyclopedia of Mathematical Sciences including its applications. Third volume in three parts: Geometry . Teubner, Leipzig 1910 ( PDF full text from the Göttingen Digitization Center [accessed on April 13, 2012]).
• Jeremy Gray: Worlds out of nothing: a course of the history of geometry of the 19th century . Springer, 2007, ISBN 978-0-85729-059-5 .

References and comments

1. ↑ ^{a b} Beutelspacher & Rosenbaum (2004)
2. ↑ In fact, the term “quadratic set” is in many cases really more comprehensive than “projective quadric” and therefore not equivalent to this analytical term. The terms in finite, Desarguese Fano levels are equivalent, see the examples in this article.
3. ^ Fano (1910), I.1
4. ↑ Benz (1990)
5. ↑ Buekenhout (1969)
6. ↑ is one such geometry throughout this article. ${\displaystyle \mathbb {P} }$
7. ↑ ^{a b} From the definition it follows that an oval or ovoid is a non-degenerate quadratic set of plane or space.
8. ↑ ^{a b} Polster (1991), 1.6 Ovals and Hyperovals
9. ↑ Beutelspacher & Rosenbaum (2004), definition p. 147.
10. ↑ Beutelspacher & Rosenbaum (2004), sentence 4.2.4
11. ↑ The sentence goes back to Ernst Witt . Beutelspacher & Rosenbaum (2004), Theorem 4.4.4
12. ↑ Beutelspacher & Rosenbaum (2004), sentence 4.3.1
13. ^ Beutelspacher & Rosenbaum (2004), sentence 4.7.4
14. ^ Beutelspacher & Rosenbaum (2004), sentence 4.7.5
15. ^ After Beniamino Segre : Sulle ovali nei piani lineari finiti . In: Atti Accad. Naz. Lincei Rendic . tape 17 , 1957, pp. 141-142 .
16. ↑ Beutelspacher & Rosenbaum (2004), sentence 4.5.1
17. ^ Beutelspacher & Rosenbaum (2004), sentence 4.5.3
18. ↑ Beutelspacher & Rosenbaum (2004), Corollary 4.5.4
19. ↑ Beutelspacher & Rosenbaum (2004), Lemma 4.4.1

Web links

• Projective geometry , short script, University of Darmstadt (PDF; 180 kB)