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✖We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.
The paper is devoted to the description of family of scalene triangles for which the triangle formed by the intersection points of bisectors with opposite sides is isosceles. We call them Sharygin triangles. It turns out that they are parametrized by an open subset of an elliptic curve. Also we prove that there are infinitely many non-similar integer Sharygin triangles. © 2017 Korean Mathematial Soiety.
We apply Orlov's equivalence to derive formulas for the Betti numbers of maximal Cohen-Macaulay modules over the cone an elliptic curve $(E,x)$ embedded into $\mathbb{P}^{n-1}$, by the full linear system $|\mathcal{O}(nx)|$, for $n>3$. The answers are given in terms of recursive sequences. These results are applied to give a criterion of (Co-)Koszulity.
In the last two sections of the paper we apply our methods to study the cases $n=1,2$\,. Geometrically these correspond to the embedding of an elliptic curve into a weighted projective space. The singularities of the corresponding cones are called minimal elliptic. They were studied by K.Saito \cite{Saito74}, where he introduced the notation $\widetilde{E_8}$ for $n=1$, $\widetilde{E_7}$ for $n=2$ and $\widetilde{E_6}$ for the cone over a smooth cubic, that is, for the case $n=3$\,. For the singularities $\widetilde{E_7}$ and $\widetilde{E_8}$ we obtain formulas for the Betti numbers and the numerical invariants of MCM modules analogous to the case of a plane cubic.
We show that if we are given a smooth non-isotrivial family of curves of genus 1 over C with a smooth base B for which the general fiber of the mapping J : B → A 1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on H1 (·, Z) of the fibers) coincides with SL(2, Z); if the general fiber has m > 2 connected components, then the monodromy group has index at most 2m in SL(2, Z). By contrast, in any family of hyperelliptic curves of genus g > 3, the monodromy group is strictly less than Sp(2g, Z). Some applications are given, including that to monodromy of hyperplane sections of Del Pezzo surfaces.