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PARALLELISMS OF PROJECTIVE SPACES

PARALLELISMS OF PROJECTIVE SPACES. Vorrei e Mi dispiace:. Vorrei ringraziare il commitato scientifico per fare l’organizazione di questo bel convegno--- Sarebbe piacevole se potrei parlare in italiano ma parlo nel modo quasi-brutto quindi

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PARALLELISMS OF PROJECTIVE SPACES

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  1. PARALLELISMS OF PROJECTIVE SPACES

  2. Vorrei e Mi dispiace: Vorrei ringraziare il commitato scientifico per fare l’organizazione di questo bel convegno--- Sarebbe piacevole se potrei parlare in italiano ma parlo nel modo quasi-brutto quindi mi dispiace-----non voglio provacare u’incidente internazionale ---

  3. What is a Line Spread? ----A covering of the points by lines-----

  4. And, a Parallelism? • Well,-----this is a set of spreads that share no lines but cover the line set----

  5. What are the Main Problems? -----Existence and Construction coupled with characterization theorems-----

  6. Parallelisms in PG(3,K). Considering parallelisms in PG(3,K), for K a skewfield, a spread corresponds to a translation plane -----so the theory of this area can be utilized ----

  7. Main Problem on Translation Planes and Parallelisms. Given a translation plane with spread in PG(3,K), either determine a parallelism containing it or show that no such parallelism can exist.

  8. Diversity would be nice! Perhaps the most challenging problem would be to find a parallelism such that all associated translation planes are mutually non-isomorphic.

  9. The Minimal Situation: All Spreads are Isomorphic. Given a spread S, does there exist a parallelism containing it such that all spreads are isomorphic to S?

  10. Try Something Easier! • Given a Desarguesian spread, does there exist a parallelism containing it such that all spreads are Desarguesian ----but here require that the space is finite. • ----This problem eluded mathematicians for at least twenty years ----

  11. Walker –Lunardon Theorem Desarguesian parallelisms (all spreads Desarguesian) in PG(3,q) correspond to spreads in PG(7,q) which are unions of derivable partial spreads containing a (small) regulus.

  12. Long-Standing Conjecture: ----Desararguesian (regular) Parallelisms do not exist in PG(3,q), for q odd----

  13. The Parallelisms of Prince in PG(3,5) Whoops! ---Prince found 45 “cyclic parallelisms” in PG(3,5) of which two are Desarguesian. ---A cyclic parallelism is one that admits an cyclic and transitive automorphism autotopism---

  14. The Penttila and Williams Parallelisms. Penttila and Williams found a class of class of regular parallelisms in PG(3,q), where q is congruent to 2 mod 3 that contain all previously known regular parallelisms.

  15. Recognition of Cyclic and Regular Parallelism. So, working from the associated translation plane of order q^4, there is a group isomorphic to SL(2,q)xC, where C is cyclic of order 1+q+q^2.

  16. Jha-Johnson SL(2,q)xC acting on a translation plane of order q^4 produces an associated cyclic and Desarguesian parallelism in PG(3,q)

  17. Open Problem: ----So, determine if there are regular parallelisms in PG(3,q), for q congruent to 0 or 1 mod 3---

  18. P-Primitive and Transitive Parallelisms –the Group. • A “transitive parallelism” in PG(3,q) admits a transitive group G whose order is divisible by 1+q+q^2 =(q^3 –1)/(q-1). • A “p-primitive parallelism is one that admits a collineation whose order is a p-primitive divisor of (q^3-1)/(q-1).

  19. Biliotti-Jha-Johnson • The full group is completely determined. • Usually, the group is solvable, but there are some notable exceptions--- • Conference Homework: Look at Theorem 33 in the lecture notes and, show ---before Sunday ---that the only non-solvable possibility is PSL(2,7) and q =2.

  20. Lorimer-Rahilly and Johnson-Walker The translation planes of order 16 are completely determined----there are exactly two that admit PSL(2,7) as a collineation group: L-R and J-W. These planes also admit SL(2,2)xC where C is cyclic of order 1+2+2^2. Since all spreads in PG(2,2) are Desarguesian---there are corresponding Desarguesian parallelisms.

  21. Bonus!! • Arguing from the L-R or J-W translation planes of order 16, there is a “two-transitive” parallelism. • Are there other two-transitive parallelisms? • We might look at the classification theorem of finite simple groups. • We might also look directly at the theorem of Biliotti-Jha-Johnson.

  22. Johnson---and the big hammer. • Using the classification theorem, the only two-transitive parallelisms in PG(3,q) are the two regular ones in PG(3,2). • Where can a I find a smaller hammer? • O.k. ---use Biliotti-Jha-Johnson

  23. Biliotti-Jha-Johnson’s little hammer The group is in GammalL(4,q), and we know the group ---duh! So, we get the two-transitive result “senza.”

  24. Open Problem on Transitive Parallelisms. • Completely determine all parallelisms that admit transitive groups. • Too hard! • O.k. how about talking about partial parallelisms with say 1+q+q^2-t spreads (deficiency t)? Determine the transitive t-deficient parallelisms. • Maybe we could try this for t=1; deficiency one. Here we note that there is a unique extension to a parallelism. • Before I try this, maybe I should ask if there are examples.

  25. Denniston-Parallelisms. • By looking at the associated question on the Klein quadric, Denniston constructed a class (two actually) of parallelisms containing a unique Desarguesian spread and all the rest Hall. • But, what does this have to do this transitive deficiency one? • Nothing! ---So far.

  26. Beutelspacher’s Class of Examples. • Beutelspacher constructed a class similar to that of Denniston---are the two classes equal? • Still ----what does this have to do with transitive deficiency one parallelisms?

  27. Johnson’s Construction in PG(3,K), K a field. • In PG(3,K), for K a field admitting a quadratic extension, there is a construction of transitive deficiency one parallelisms using the full central collineation group with fixed axis of a associated Desarguesian spread. • The construction provides a count of isomorphism classes of such parallelisms; there are lots of these things! • In the finite case, are any of these Denniston or Beutelspacher? • Probably ----Maratea Homework #2 ---try to prove this!

  28. Desarguesian Deficiency One • Given transitive deficiency one partial parallelism (t-d-1), there is a unique extension to a parallelism and the group acts as a collineation group of the adjoined spread. • If all spreads of a t-d-1 are Desarguesian, it must be the case the adjoined spread is Desarguesian. • Homework #3 ---show the adjointed spread is Desarguesian.

  29. Johnson’s add-on Theorem • If there is a Desarguesian deficiency one parallelism, there is an associated translation plane in PG(7,q). • The associated translation plane admits a particular net D such that the adjoined spread is Desarguesian if and only if the net D is derivable. • Is there a group characterization ---to highlight the net in question? • Yep!

  30. Johnson’s-SL-characterization theorem. • Suppose that A is an affine plane of order q^4 with spread in PG(7,q) admitting SL(2,q) whose line (component) orbits are 1 or q^2-q. • If p^r =q and the p-elements are elations ---what happens? • You get a Desarguesian parallelism. • If the p-elements are all Baer ---what happens?

  31. Eureka and Gadzooks! • This is the right question ---you get a Desarguesian deficiency one and a designated net fixed linewise ---is it derivable?? • What do I get if I can prove it is? • The Maratea Medal in Finite Geometry. • To be conferred in 3002---just to make sure the proof holds up.

  32. Back to the Transitive Question • So, maybe I could prove that transitive deficiency are Denniston/Beutelspacher/Johnson? • Well, unfortunately not! • How far off would I be? • Pretty far!

  33. Johnson’s Derived-Knuth and Derived Derived-Knuth Types • Using “some” but not all of a central collineation group of an associated Desarguesian spread ---many new examples can be constructed. • Some have one Desarguesian spread and the rest “derived” Knuth semifield planes. These are “transitive.” • Others have one Knuth semifield spread, one Desarguesian spread and the rest are derived Knuth ---but we lose transitivity.

  34. Conical Flocks and Transitive Deficiency one • If there is a transitive deficiency one how does the group act on the adjoined spread? • Good question but really the wrong direction. The best question is to ask how big the group must be and then discuss the stabilizer subgroup of one of the other spreads. • I don’t get it ---but the above title suggests that there is a connection to flocks of quadratic cones ---very strange!

  35. Johnson’s Baer groups must live in Payne-Thas Hyper-space. • A Baer group of order q acting on a spread in PG(3,q) is equivalent to a partial flock of a quadratic cone of deficiency one. • There is an extension to a flock if and only if the plane is derivable. • Payne and Thas have shown there is always an extension. • So? • Bingo! ----This is a derived translation of a plane corresponding to a quadratic cone.

  36. Conjecture on T-d-1’s. • So, given a t-d-1, if we could show that somehow, in the stabilizer of a 2nd spread there is a Baer group then all of the spreads other than the adjoined are derived conical flock spreads. • So, the conjecture is to prove this for t-d-1’s and to show that the adjoined spread is Desarguesian. • Is this Maratea homework #5? • Yep!

  37. In Fine and Generalizations • All of these problems can be generalized from PG(3,q) to parallelisms and their generalizations in higher dimensional projective spaces. • Please work on this problems. • Do you mean I should solve the general problems during this conference. • Of course!

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