By Buekenhout F., Huybrechts C.

We turn out the lifestyles of a rank 3 geometry admitting the Hall-Janko staff J2 as flag-transitive automorphism crew and Aut(J2) as complete automorphism crew. This geometry belongs to the diagram (c·L*) and its nontrivial residues are entire graphs of dimension 10 and twin Hermitian unitals of order three.

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**Extra info for A (c. L* )-Geometry for the Sporadic Group J2**

**Example text**

Proof: Suppose that P = F ⊕ Q and G is a face of P . Any element g of G can be written uniquely as f0 + q0 , where f0 ∈ F and q0 ∈ Q. Since G is a face, f0 and g0 still belong to G. Now if p1 and p2 are elements of P whose sum belongs to F + G it follows that we can write p1 + p2 = f + g0 , where f ∈ F and g0 ∈ G ∩ Q. If we write pi = fi + qi with fi ∈ F and qi ∈ Q, we see that f = f1 + f2 and g0 = q1 + q2 . It follows that each qi belongs to G and hence that each pi belongs to F + G. Thus F + G is a face of P , and the implication of (2) by (1) is proved.

Then there is a constant c ∈ R such that for all r ∈ R, #Bh (r) < crd . 1), H(Q) is finitely generated and sharp, and hence it has a unique set of minimal generators {h1 , . . hm }. 7) shows that each hi belongs to the face generated by h. 2) implies that for each i there exists an integer ni such that ni h ≥ hi in H(Q). Choose n ≥ ni for all i. Then for every r ∈ R+ , Bh (r) ⊆ ∩i Bhi (nr). 1) implies that H(Q)gp ∼ = Hom(Qgp , Z), and consequently gp {hi } spans Hom(Q , Z). 7 says that this group has rank d.

Indeed, by definition, an element h of H(Q) belongs to its interior if and only if it is not contained in any proper face of Q. , if and only if h⊥ = Q∗ . This is exactly the condition that h: Q → N be a local homomorphism. We shall find the following crude finiteness result useful. More precise variants are available, most of which rely on the theory of Hilbert polynomials in algebraic geometry. 8 Let Q be a fine sharp monoid of dimension d and let h: Q → N be a local homomorphism. For each real number r, let Bh (r) := {q ∈ Q : h(q) < r}.