## Projective planes in algebraically closed fields (1991)

Venue: | Proc. London Math. Soc |

Citations: | 4 - 0 self |

### BibTeX

@TECHREPORT{Evans91projectiveplanes,

author = {David M. Evans and Ehud Hrushovski},

title = {Projective planes in algebraically closed fields},

institution = {Proc. London Math. Soc},

year = {1991}

}

### Years of Citing Articles

### OpenURL

### Abstract

We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. This is applied to give new examples of algebraic characteristic sets of matroids. The main technique used, which is motivated by classical projective geometry, is that a particular configuration of four lines and six points in the geometry indicates the presence of a connected one-dimensional algebraic group.

### Citations

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Citation Context ..., it is isomorphic to the additive group of L [8, Theorem 20.5]. If, on the other hand, G is complete, then it is commutative, and is isomorphic to an elliptic curve defined over K (see, for example, =-=[20]-=- for definitions). Thus, in any case G is commutative, and it follows from the above that if H, G are two connected one-dimensional algebraic groups, then Hom(//, G) is non-zero only if H is isomorphi... |

333 |
Matroid Theory
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Citation Context ...ation x ~y if and only if c\K({x}) = c\K({y}) on L — K. We use the same notation for the pregeometry and the geometry, the usage hopefully being clear from the context. The reader might consult, say, =-=[24]-=- for general background on pregeometries. A.M.S. (1980) subject classification: 03C60, 05B35. Proc. London Math. Soc. (3) 62 (1991) 1-24.s2 DAVID M. EVANS AND EHUD HRUSHOVSKI It is convenient to have ... |

221 | Linear Algebraic Groups - Humphreys - 1975 |

66 |
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Citation Context ...as.) We now introduce some terminology from algebraic geometry, suitably adapted for our purposes. The presentation is necessarily sketchy: for fuller accounts the reader should consult, say, [22] or =-=[17]-=-. For any neN, a subset V of L n is called a Zariski closed set if there exists a (finite) set of polynomials {Pi(Xlt..., Xn): i el} inn variables with coefficients in K such that V = {(xlt..., xn) e ... |

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Citation Context ...lt ..., as.) We now introduce some terminology from algebraic geometry, suitably adapted for our purposes. The presentation is necessarily sketchy: for fuller accounts the reader should consult, say, =-=[22]-=- or [17]. For any neN, a subset V of L n is called a Zariski closed set if there exists a (finite) set of polynomials {Pi(Xlt..., Xn): i el} inn variables with coefficients in K such that V = {(xlt...... |

37 |
Algebraic Number Fields
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Citation Context ... U ({primes p: f does not split into linear factors modulo p} D xL(M<t>))- REMARK. If / is monic, then the density of the set of primes in the above is given by the Frobenius-Tchebotarev theorem (see =-=[10]-=-, for example). It is interesting to speculate on just what the possibilities for Xs or XA can be. Clearly there can only be countably many such sets, and in fact it is not difficult to convince onese... |

28 |
Linear algebra and projective geometry
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Citation Context ...ndence. We say that such a geometry is a projective geometry over D, or that it is coordinated by D. If the dimension of a projective geometry is at least 4, then classical results (see, for example, =-=[2]-=- or [21]) show that it is coordinatised by some skew-field. If the dimension is 3, then we refer to the geometry as a projective plane. A projective plane need not be coordinatised by a skew-field. In... |

26 |
Theorie de Galois Imaginaire
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Citation Context ...if and only if [(a = b) or (for t e T with t d« a"b we have <f>(t, a) and <fr(t, b) defined and equal)]. As above, this is an equivalence relation, which clearly agrees with ~ on 5. By The"oreme 7 of =-=[18]-=-, it is enough to show that =» is constructible, that is, the condition that a «b can be expressed using a finite number of ^-polynomial equations and inequations between the components of a and b. As... |

12 |
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Citation Context ...quasivarieties, quasimorphisms and quasialgebraic groups (cf. [17] for more details). We shall need the following result, which is an observation by the second author (in [7]) that results of Weil in =-=[23]-=- can be generalised to non-zero characteristic (see (5.23) and (4.13) in [17] for details, and also see [4]). THEOREM 1.2.1. Let S be a T-orbit on L n for some n. Suppose there exists a locally quasir... |

12 |
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Citation Context ... there exists 6 e T fixing b such that ) = y2/>, whence dgx = g2, as required. The above also shows that r(/f) = iRemark. Results (d) and (e) above are almost immediate consequences of Theorem 3.1 of =-=[25]-=-.) To summarise, H is a closed subgroup of G x G which is connected and one-dimensional, the projections Kt: H—>G (which are morphisms) are surjective and have finite kernel. Moreover, if {tu t2) is a... |

10 |
Some Interpretations of Abstract Linear Dependence in terms of Projective Geometry
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(Show Context)
Citation Context ...of V£mos), and matroids where XA and Xs contain infinitely many and omit infinitely many primes. The technique for producing the examples of Xs is really little more than the observation (implicit in =-=[16]-=-) that we can mimic the condition of satisfiability of a set of equations in a skew-field by the linear representability of a particular matroid over that skew-field. The matroids we produce can easil... |

7 |
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Citation Context ..., u) for at least m values of i =s r O 0(r, u) = <f)(t, v) for some t e T with t dj iTu This proves (ii). Finally, (iii) is just a consequence of Th6oreme 7 of [18]. COROLLARY 1.2.3 (Dress and Lovasz =-=[3]-=-). Let U, W be algebraically closed subfields of L which contain K. Then there is a subfield TofU {containing K) such that for any subfield S of U, we have Ud>sW if arid only if S contains T. Proof. A... |

7 | Homogeneous geometries
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(Show Context)
Citation Context ... r(jc). Then <j> induces a ring homomorphism from K[a"xJ] onto K[a'~x'y], which by Theorem II.1.3 of [22] is an isomorphism. This isomorphism extends to an automorphism of L. Finally, note that a' = 0=-=(5)-=- lies in tf>(cl(B U C)) = c\(B) = B, and similarly x' lies in Z. THEOREM 2.1.2. Let (A, B, C, X, Y, Z) be a partial quadrangle in G(L/K). Then there exist an irreducible one-dimensional algebraic grou... |

4 |
Non-algebraic matroids exist
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Citation Context ...y that any line parallel to two parallel lines lies in the plane containing them (two lines are parallel if they are coplanar and disjoint). That the geometries G(L/K) have this property was shown in =-=[9]-=-, and in fact it follows easily from Corollary 1.2.3. Perhaps there is such a 'geometric' proof. However, we now show that the conclusion of Corollary 3.2.1 implies that any connected one-dimensional ... |

4 |
Characteristic sets of matroids
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Citation Context ...n over a field of characteristic p. X3)s18 DAVID M. EVANS AND EHUD HRUSHOVSKI The problem of determining the possibilities for the linear characteristic set of a matroid is now completely solved (see =-=[11]-=-), and the answer is straightforward: if 0 e XL(M) then all but finitely many primes are in XL(M), and any finite set of primes can be omitted; if Q$xdM) (or indeed if 0$Xs(M)) then (as was first obse... |

3 |
A Desarguesian theorem for algebraic combinatorial geometries
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(Show Context)
Citation Context ... unfamiliar geometry is to look for subgeometries which one can understand. Here, we look at subgeometries of G(L/K) which are projective planes (see § 3.3 for definitions). As was already known (cf. =-=[13]-=-), any projective plane arising as a subgeometry of G(L/K) is Desarguesian, and so is isomorphic to a projective plane coming from a 3-dimensional vector space over a skew-field. The main new result (... |

2 |
Intersections of algebraically closed fields
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(Show Context)
Citation Context ...ebraic techniques can be used to prove some of the results in this paper: for example, Corollary 3.2.1 (but see Remark 2 of §4.1). In characteristic zero, derivations will suffice (cf. Theorem 4.3 of =-=[1]-=-). However, in non-zero characteristic, what seems to be lacking is a way of computing intersections of algebraically closed fields which is of comparable power to Theorem 1.5 of [1]. Acknowledgements... |

2 |
On P-Polynomial Representations of Projective Geometries
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(Show Context)
Citation Context ...'e K[T] (a p-polynomial), where p is the characteristic of K, such that, for all x e L, <t>(x) = E,sn (XiX p> . The p-polynomials in K[T] form a semigroup under composition, and it is well-known (see =-=[15]-=- for a proof) that this semigroup satisfies the condition in Fact 2.2.1 (the left Ore condition). (ii) G, H are isomorphic to Gm(L), the multiplicative group of L. In this case Hom(G, #) = Z, as any e... |

1 |
Groups in characteristic zero are algebraic groups
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(Show Context)
Citation Context ...owing result, which is an observation by the second author (in [7]) that results of Weil in [23] can be generalised to non-zero characteristic (see (5.23) and (4.13) in [17] for details, and also see =-=[4]-=-). THEOREM 1.2.1. Let S be a T-orbit on L n for some n. Suppose there exists a locally quasirational map m which is well-defined on pairs of independent elements of S and which satisfies the following... |

1 |
The WWW computer architecture maintained by Milo Martin http://www.cs.wisc.edu/arch/www
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Citation Context ...G) = Hom(G, G) can be considered as a ring. Then Fact 2.2.1 says precisely that E(G) is a left Ore domain. Thus E(G) embeds in a 'skew field of quotients', which we denote by E°(G) (see, for example, =-=[6]-=-). We list the possibilities for E°(G). Case 1: p = 0. (i) If G = Ga(L) then E°(G) = K. (ii) If G = GW(L) then £°(G) = Q, the rationals. (iii) If G is an elliptic curve, then E°(G) is isomorphic to Q ... |

1 | Introduction to algebraic geometry (Wiley-Interscience - LANG - 1958 |

1 |
Matroids, algebraic and non-algebraic', Algebraic, extremal and metric combinatorics
- LINDSTROM
- 1986
(Show Context)
Citation Context ...L/K). The motivation for this is twofold. Firstly, a certain amount of work has already been done on these geometries by combinatorialists working in matroid theory (cf. for example, the survey paper =-=[14]-=-). Secondly, these geometries are important in model theory, and are central to a conjecture of Zil'ber on ^-categorical theories (see [26] for a survey). Although this conjecture is now known to be f... |

1 |
Classification theory (North
- SHELAH
- 1987
(Show Context)
Citation Context ...base of the type of W over U (see [19, III.6.10]). Elimination of imaginaries in L means that we may think of this canonical base as lying in U, and the statement now follows from Theorem III.6.10 of =-=[19]-=-. 2. Theorems 2.1. Groups from partial quadrangles Let K and L be algebraically closed fields as in the previous section. We shall be considering the (pre)geometry G(L/K) on L given by algebraic closu... |

1 |
Projektive Geometrie (Bibliographisches Institut
- TAMASCHKE
- 1969
(Show Context)
Citation Context ... We say that such a geometry is a projective geometry over D, or that it is coordinated by D. If the dimension of a projective geometry is at least 4, then classical results (see, for example, [2] or =-=[21]-=-) show that it is coordinatised by some skew-field. If the dimension is 3, then we refer to the geometry as a projective plane. A projective plane need not be coordinatised by a skew-field. In fact, i... |

1 |
Towards the structural stability theory', preprint
- ZIL'BER
- 1987
(Show Context)
Citation Context ...working in matroid theory (cf. for example, the survey paper [14]). Secondly, these geometries are important in model theory, and are central to a conjecture of Zil'ber on ^-categorical theories (see =-=[26]-=- for a survey). Although this conjecture is now known to be false, a better understanding of the geometries is still desirable. One way of trying to understand an unfamiliar geometry is to look for su... |