Algebraic k -systems of curves
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Algebraic k-systems of curves Charles Daly1 · Jonah Gaster2 · Max Lahn3 · Aisha Mechery4 · Simran Nayak5 Received: 20 November 2019 / Accepted: 10 March 2020 © Springer Nature B.V. 2020
Abstract A collection of simple closed curves on an orientable surface is an algebraic k-system if the algebraic intersection number α, β is equal to k in absolute value for every α, β ∈ . Generalizing a theorem of Malestein et al. (Geom Dedicata 168(1):221–233, 2014. doi:10.1007/s10711-012-9827-9) we compute that the maximum size of an algebraic k-system of curves on a surface of genus g is 2g +1 when g ≥ 3 or k is odd, and 2g otherwise. To illustrate the tightness in our assumptions, we present a construction of curves pairwise geometrically intersecting twice whose size grows as g 2 . Keywords Curves on surfaces · k-Systems · Topological configurations Mathematics Subject Classification 57K20
1 Introduction Questions about the combinatorial properties of collections of curves on surfaces with specified pairwise intersection data have been considered by various authors. In [6], the first explicit bounds were obtained for the maximum size of any curve collection on a surface of
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Jonah Gaster [email protected] Charles Daly [email protected] Max Lahn [email protected] Aisha Mechery [email protected] Simran Nayak [email protected]
1
Department of Mathematics, University of Maryland, College Park, USA
2
Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, USA
3
Department of Mathematics, University of Michigan, Ann Arbor, USA
4
Department of Mathematics, Bryn Mawr College, Bryn Mawr, USA
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Department of Applied Mathematics, Brown University, Providence, USA
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Geometriae Dedicata
finite-type such that any pair has geometric intersection number at most k. In the remarkably delicate case k = 1, increasingly precise bounds on these maximum sizes are given in [1], [4], [7], and [8], but aside from several low-complexity examples, exact values remain unknown. In [7] it is shown that the maximum size of a collection of simple curves on a connected oriented surface S, of genus g, such that every pair of distinct curves intersects an odd number of times is 2g+1. The authors’ proof exploits the symplectic pairing on H1 (S; Z/2Z) induced by the algebraic intersection number. Our main theorem is an extension of this result. The proof is slightly different than that of [7], using the symplectic pairing of vectors to apply induction on g. This allows a bit more information. We say that a subset ⊂ Z2g is a symplectic k-system if |v, w| = k for all distinct v, w ∈ . Theorem 1 Let k ≡ 2m−1 (mod 2m ). A symplectic k-system has size at most 2g + 1, and equality is realized. When g ≥ 3 or m = 1, there exist primitive symplectic k-systems of size 2g + 1. When m > 1 and g ≤ 2, a primitive symplectic k-system has size at most 2g, and equality is realized. Since the geometric and algebraic intersection numbers of any pair of curves on S have the same parity, [7, Thm 1.4] is the case m = 1 above. Na
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