Geometry on the lines of spine spaces
Krzysztof Petelczyc [email protected] Mariusz Zynel˙ [email protected]
University of Bialystok Institute of Mathematics
XI P´o lnocne Spotkania Geometryczne, Gda´nsk 2017 Our goal
To recover the pointset of a spine space from the set of its lines using a binary relation π or ρ.
K. Petelczyc, M. Zynel˙ Geometry on the lines of spine spaces, sent to Aequationes Math.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Motivations and references
M. Pieri Sui principi che regono la geometria delle rette Atti Accad. Torino 36 (1901), 335–350. K. Pra˙zmowski, M. Zynel˙ Possible primitive notions for geometry of spine spaces J. Appl. Logic 8 (2010), no. 3, 262–276. W.-L. Chow On the geometry of algebraic homogeneous spaces Ann. of Math. 50 (1949), 32—67. A. Kreuzer Locally projective spaces which satisfy the Bundle Theorem J. Geom. 56 (1996), 87–98 K. Petelczyc, M. Zynel˙ Coplanarity of lines in projective and polar Grassmann spaces Aequationes Math. 90 (2016), no. 3, 607–623.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Grassmann spaces
V – a vector space of dimension n with 3 ≤ n < ∞
Subk (V ) – the set of all k-dimensional subspaces of V Assume that 0 < k < n.
For H ∈ Subk−1(V ), B ∈ Subk+1(V ) with H ⊂ B a k-pencil is a set of the form
p(H, B) := [H, B]k = U ∈ Subk (V ): H ⊂ U ⊂ B . The point-line structure
Pk (V )= Subk (V ), Pk (V ) , where Pk (V ) is the family of all k-pencils, is a Grassmann space.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Spine spaces
W – a fixed subspace of V m – an integer such that k − codim(W ) ≤ m ≤ k, dim(W )
Fk,m(W ) := {U ∈ Subk (V ): dim(U ∩ W )= m}
Gk,m(W ) := {L ∩ Fk,m(W ): L ∈ Pk (V ) and |L ∩ Fk,m(W )|≥ 2}
The point-line structure
M = Ak,m(V , W )= Fk,m(W ), Gk,m(W ) is called a spine space.
M is a Gamma space M can be a projective space a slit space an affine space the space of linear complements
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Lines in spine spaces
M is a fragment of the Grassmann space A – affine lines Lα and Lω – two types of projective lines L := A∪Lα ∪Lω
∞ class representative line g = p(H, B) ∩Fk,m(W ) g
Ak,m(W ) H ∈Fk−1,m(W ), B ∈Fk+1,m+1(W ) H + (B ∩ W ) α Lk,m(W ) H ∈Fk−1,m(W ), B ∈Fk+1,m(W )– ω Lk,m(W ) H ∈Fk−1,m−1(W ), B ∈Fk+1,m+1(W )–
Table: The classification of lines in a spine space Ak,m(V , W ).
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Maximal strong subspaces
Grassmann space Pk (V ): projective
stars S(H) = [H, V ]k = {U ∈ Subk (V ): H ⊂ U}, where H ∈ Subk−1(V )
tops T(B)=[Θ, B]k = {U ∈ Subk (V ): U ⊂ B}, where B ∈ Subk+1(V )
Spine space Ak,m(V , W ): projective and semiaffine (a projective space P with a subspace D removed)
class representative subspace dim(P) dim(D)
ω-stars [H, H + W ]k : H ∈ Fk−1,m−1(W ) dim(W ) − m -1
α-stars [H, V ]k ∩ Fk,m(W ): H ∈ Fk−1,m(W ) dim(V ) − k dim(W ) − m − 1
α-tops [B ∩ W , B]k : B ∈ Fk+1,m(W ) k − m -1
ω-tops [Θ, B]k ∩ Fk,m(W ): B ∈ Fk+1,m+1(W ) k k − m − 1
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Maximal strong subspaces in spine spaces
two stars or two tops: disjoint, share a point a projective star and a projective top: disjoint, share a point (in remaining cases) a star and a top: disjoint, share a line
Fact A line of M can be in at most two maximal strong subspaces of different type: a star and a top.
Lemma Three pairwise coplanar and concurrent, or parallel, lines not all on a plane span a star or a top.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Two binary relations on lines...
E – plane in M, U ∈ E U2 p(U, E) := L ∈L: U ∈ L ⊆ E E U1 is a pencil of lines if U is proper U3 U is a parallel pencil otherwise coplanarity
L1 π L2 iff there is a plane E such that L1, L2 ⊂ E
relation of being in one pencil of lines
L1 ρ L2 iff there is a pencil p such that L1, L2 ∈ p
ρ ⊆ π
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces ...and their cliques...
X – a subspace of M
L(X )= {L ∈L: L ⊂ X }
flat: L(E), where E is a plane semiflat: all projective lines and “a selector” on a plane flats and semiflats can be: projective, punctured, affine (punctured ∪ affine = semiaffine)
U ∈ X LU (X )= {L ∈L: U ∈ L and L ⊆ X }
semibundle: LU (X ), where X is a strong subspace of M semibundles can be: proper or improper
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces ... maximal in particular
Lemma
1 Flats and semibundles are π-cliques. 2 Semiflats and proper semibundles are ρ-cliques.
Proposition Every maximal π-clique is either a flat or a semibundle.
Proposition Every maximal ρ-clique is either a semiflat or a proper semibundle.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Cliques in terms of δ ∈{π, ρ}
∆δ(L1, L2, L3) iff 6= (L1, L2, L3) and Li δ Lj for all i, j =1, 2, 3,
and for all M1, M2 ∈ L if M1, M2 δ L1, L2, L3 then M1 δ M2.
Lemma
Let L1, L2, L3 ∈ L.
1 ∆π(L1, L2, L3) iff L1, L2, L3 form a tripod or a triangle.
2 ∆ρ(L1, L2, L3) iff L1, L2, L3 form a ρ-clique, they are not in a pencil of lines, they are not on an affine plane, and in case they are on a punctured plane one of L1, L2, L3 is an affine line.
generally speaking: 3 lines satisfying ∆δ determine a δ-clique “more than a plane” assumption: 3 ≤ n − k and 3 ≤ k − m
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Maximal cliques in terms of δ ∈{π, ρ}
The set
[|L1, L2, L3|]δ := L ∈ L: L δ L1, L2, L3
is the maximal δ-clique, provided that ∆δ(L1, L2, L3). Proposition
1 The family of maximal π-cliques is definable in hL, πi.
2 Maximal ρ-cliques, except affine semiflats, are definable in hL, ρi.
Lemma A maximal ρ-clique K satisfies the following condition:
there are lines L1 ∈ K, L2 ∈L\ K such that
(K \{L1}) ∪{L2} is a maximal ρ-clique (∗)
iff K is a semiaffine semiflat.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Ternary concurrency
Let L1, L2, L3 ∈L.
pπ(L1, L2, L3) iff Li π Lj for all i, j = 1, 2, 3 and ¬∆π(L1, L2, L3)
iff L1, L2, L3 form a pencil of lines or a parallel pencil
pρ(L1, L2, L3) iff there are M1, M2, M3 ∈L such that
∆ρ(M1, M2, M3), [|M1, M2, M3|]ρ does not satisfy (∗),
L1, L2, L3 ∈ [|M1, M2, M3|]ρ, and ¬∆ρ(L1, L2, L3)
iff L1, L2, L3 form a pencil of lines
Lemma
1 The family Pπ of all pencils of lines and parallel pencils is definable in hL, πi.
2 The family Pρ of all pencils of lines is definable in hL, ρi.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Parallel pencils
Coplanar pencils in hL, πi
p1 Π p2 iff for all l1 ∈ p1, l2 ∈ p2 we have l1 π l2
Parallel pencils in hL, πi
on an affine plane: p1 is a parallel pencil if there is another pencil p2 such that p1 Π p2 and p1 ∩ p2 = ∅ on a punctured plane: p is a parallel pencil if the base plane of p is not affine but every line l ∈ p lies on some affine plane
Lemma
The family Pk of all parallel pencils is definable in hL, πi.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Pencils of lines
P – the family of all pencils of lines in M
for π: P = Pπ \ Pk for ρ: P = Pρ
Proposition If M satisfies “more than a plane” assumption, then hL, Pi is definable in hL, δi.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Geometry induced by pencils of lines, proper semibundles
Proper maximal δ-cliques: projective flats punctured semiflats proper semibundles “more than a 3-space” assumption: stars or tops are at least 4-dimensional projective or semiaffine spaces
P0 – the family of all pencils of lines definable in hL, δi K0 K δ := K ∈ δ : there is q ∈ P0 such that q ⊂ K B K0 := K ∈ δ : dim(K) ≥ 3 Lemma The family B coincides with the family of all proper top semibundles, the family of all proper star semibundles or the union of these two families depending on whether tops, stars or all of them as projective or semiaffine spaces are at least 4-dimensional.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Grouping proper semibundles into bundles
B Let Ki := LUi (Xi ) ∈ , i = 1, 2.
Υ(K1, K2) iff (∃ L1, L2 ∈ K1)(∃ M1, M2 ∈ K2)
L1 6= L2 ∧ L1 δ M1 ∧ L2 δ M2 Lemma
If Υ(K1, K2) and K1 ∩ K2 = ∅, then X1, X2 are both stars or tops and U1 = U2.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Why punctured spaces are bad?
U – a fixed point
X1 – a semiaffine, but not affine, star containing U; an α-star
X2 – a projective star containing U; an ω-star
Ki = LU (Xi ), i = 1, 2 – proper semibundles
L – a projective line in K1; an α-line
suppose that there is a line M in K2 coplanar with L the plane E spanned by L, M is contained in some α-top
there is no line in E ∩ X2, a contradiction
there is no line in K2 coplanar with L if L is an affine line, then E is contained in an ω-top
to have Υ(K1, K2) we need at least two affine lines in K1
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Grouping proper semibundles into bundles cont.
Assume that: stars or tops are at least 4-dimensional projective or semiaffine but not punctured projective spaces:
4 ≤ n − k and dim(W ) 6= m + 1 or 4 ≤ k − m and k 6= m + 1
Lemma
If X1, X2 are both stars or tops and U1 = U2, then Υ(K1, K2) and K1 ∩ K2 = ∅.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Bundles of lines
Υ∅(K1, K2) iff Υ(K1, K2), Υ(K2, K1), and either K1 ∩ K2 = ∅ or K1 = K2
Lemma
Υ∅(K1, K2) iff X1, X2 are both stars or tops and U1 = U2.
For K ∈ B we write ′ B ′ ΛΥ∅ (K) := K ∈ : Υ∅(K, K ) . [ Lemma If X is a maximal strong subspace containing a point U, then
ΛΥ∅ (LU (X )) = {L ∈L: U ∈ L}.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces The main theorem
Theorem Let M be a spine space and let L be its lineset. If stars or tops in M are at least 4-dimensional projective or semiaffine but not punctured projective spaces, then
the spine space M and the structure hL, δi (∗∗) are definitionally equivalent.
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Excluded cases where (∗∗) holds true
dim(W )= n M is a Grassmann space
dim(W )= m = k M is a single point
dim(W )= m = k − 1
M is a star in Pk (V ), i.e. it is a projective space
dim(W )= k + 1, m = k
M is a top in Pk (V ), i.e. it is a projective space
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Excluded cases where (∗∗) does not hold true
dim(W )= k, m = k − 1
M is the neighbourhood of a point W in Pk (V ), i.e. the set of all points that are collinear with W
X – a star in M ϕ – a homology 6= id on X with the center W ϕ(U), U ∈ X f : Subk (V ) −→ Subk (V ), f (U)= (U, U ∈/ X LS – the set of all lines contained in all stars
FS : LS −→ LS such that FS(L)= f (L)
LT – the set of all lines contained in all tops
FT : LT −→ LT such that FT (L)= L
F := FS ∪ FT is an automorphism of hL, δi which does not preserve bundles of lines
Krzysztof Petelczyc, Mariusz Zynel˙ Geometry on the lines of spine spaces Thank you for your attention