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We study the conditions under which an algebraic curve can be modeled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus over an algebraically closed field is nondegenerate in the above sense. More generally, let be the locus of nondegenerate curves inside the moduli space of curves of genus . Then we show that , except for where ; thus, a generic curve of genus is nondegenerate if and only if .
Let be an elliptic curve of conductor and let be a prime. We consider trace-compatible towers of modular points in the noncommutative division tower . Under weak assumptions, we can prove that all these points are of infinite order and determine the rank of the group they generate. Also, we use Kolyvagin’s construction of derivative classes to find explicit elements in certain Tate–Shafarevich groups.
We present a relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two. This allows one to decide easily if a given Cartan scheme of rank two admits a finite root system. We obtain obstructions and sharp bounds for the entries of the Cartan matrices.
The recent proof of the Boij–Söderberg conjectures reveals new structure about Betti diagrams of modules, giving a complete description of the cone of Betti diagrams. We begin to expand on this new structure by investigating the semigroup of such diagrams. We prove that this semigroup is finitely generated, and answer several other fundamental questions about it.