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Geometry on the Lines of Spine Spaces Geometry on the lines of spine spaces Krzysztof Petelczyc [email protected] Mariusz Zynel˙ [email protected] University of Bia lystok 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.
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