Algebraic \(k\)-systems of curves (Q2205672)

Let \(\Delta\) denote a collection of simple closed curves on an orientable surface \(S\). For every \(u, v \in \Delta\) if the algebraic intersection number \(\left\langle u, v \right\rangle\) is equal to \(k\) in absolute value then \(\Delta\) is called an algebraic \(k\)-system. A subset \(\Lambda \subset \mathbb{Z}^{2g}\) is called a symplectic \(k\)-system if \(\vert\left\langle u , v \right\rangle\vert = k\) for all distinct \(u,v \in \Lambda\). Let \(k \equiv 2^{m-1}\) (\(mod\,\, 2^m\)). In this paper, the authors prove that a symplectic \(k\)-system has size at most \(2g +1\), and equality is achieved. If the genus \(g \geq 3\) or power \(m=1\), there exist primitive symplectic \(k\)-systems of size \(2g + 1\). If \(m > 1\) and the genus \(g \leq 2\), a primitive symplectic \(k\)-system has size at most \(2g\), and equality is achieved. This paper also contains three constructions. The first one builds a large algebraic \(k\)-system of simple curves \(\Delta_0\). Any pair of distinct curves in \(\Delta_0\) algebraically intersect \(\pm k\) times, actually the geometric intersection of them is \(k\) in this construction, and they determine \(\mathbb{Z}\)-homology classes with symplectic pairing \(\pm k\). If \(k\) is odd, that is, \(m = 1\), one more curve can be added to \(\Delta_0\). If \(m > 1\), this curve is not primitive. The second construction gives primitive symplectic \(k\)-systems in \(\mathbb{Z}^{2g}\) of size \(2g + 1\) if the genus \(g \geq 3\) and with no restriction on \(m\). The last construction given in the paper shows that the maximum size of a collection of curves is at least quadratic in the genus so that every pair of curves geometrically intersects twice.