Algebraic k-systems of curves
Tóm tắt
A collection
$$ \Delta $$
of simple closed curves on an orientable surface is an algebraic k-system if the algebraic intersection number
$$ \langle \alpha , \beta \rangle $$
is equal to k in absolute value for every
$$ \alpha , \beta \in \Delta $$
. 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
$$g\ge 3$$
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
$$g^2$$
.
Tài liệu tham khảo
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