Let $K$ be an ample field, $G$ a finite group, and $L$ a finite Galois extension of $K$ such that $\mathrm{Gal}(L/K)$ is isomorphic to a subgroup of $G$. We prove that $K\left(x\right)$ has a Galois extension $F$ which is regular over $L$ such that $\mathrm{Gal}(F/K(x\left)\right)≌G$ and $F$ has a $K$-place $\phi$ such that $\phi \left(x\right)\in K$ and $\phi \left(F\right)=L\cup \{\infty \}$.

The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull algorithm for the Bruhat–Tits building of ${\mathrm{SL}}_d\left(K\right)$ and techniques for computing with apartments and membranes. While the original inspiration was the work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel and Tevelev in algebraic geometry, our tropical algorithms will also be applicable to problems in other fields of mathematics.

The relation between the quantum $𝒟$-modules of a smooth variety $X$ and a toric bundle is studied here. We describe the relation completely when $X$ is a semi-ample complete intersection in a toric variety. In this case, we obtain all the relations in the small quantum cohomology ring of the bundle.

Consider a root system $R$ and the corresponding toric variety $V_R$ whose fan is the Weyl fan and whose lattice of characters is given by the root lattice for $R$. We prove the vanishing of the higher cohomology groups for certain line bundles on $V_R$ by proving a purely combinatorial result for root systems. These results are related to a converse to Mazur’s Inequality for (simply-connected) split reductive groups.

For every $g\ge 3$, we show there exists a hyperelliptic curve of genus $g$ with $p$-rank $g-3$ and $a$-number $1$ in characteristic $p$ when $p=3$ or $p=5$. The method of proof is to show that a generic point of the moduli space of hyperelliptic curves of genus $3$ and $p$-rank $0$ has $a$-number $1$. When $p=3$, we also show that this moduli space is irreducible.

We study relations among the classical thetanulls of cyclic curves, namely curves $𝒳$ (of genus ) with an automorphism $\sigma$ such that $\sigma$ generates a normal subgroup of the group $G$ of automorphisms, and $g(𝒳/\langle \sigma \rangle )=0$. Relations between thetanulls and branch points of the projection are the object of much classical work, especially for hyperelliptic curves, and of recent work, in the cyclic case. We determine the curves of genus $2$ and $3$ in the locus ${\mathcal{M}}_g(G,\mathbf{C})$ for all $G$ that have a normal subgroup $\langle \sigma \rangle$ as above, and all possible signatures $\mathbf{C}$, via relations among their thetanulls.

Let $p$ be a prime integer and ${\mathbf{H}}_{\mathbf{g}}$ be a collection of complex positive definite symmetric $g\times g$ matrices $\tau$. Denote by $p\tau$ the multiplication of $\tau$ by $p$. In this note we describe an explicit process to obtain algebraic identities between theta functions with integral characteristics evaluated at $\tau$ and $p\tau$. For $g=1$ this produces modular equations between $\lambda \left(\tau \right)$, $\lambda \left(p\tau \right)$ where $\lambda \left(\tau \right)$ is the invariant associated with elliptic curve generated by $\tau$, described by the equation: $y^2=x(x-1)(x-\lambda ({\tau }_1\left)\right)$. Consequently, if the algebraic identities we obtain might serve as a higher dimensional generalization for the one dimensional modular equations.

We use term regular directed graph (r. d. g.) for the graph of irreflexive binary relation with the constant number outputs (or inputs) for each vertex. The paper is devoted to studies of maximal size $E_R(d,v)$ of r. d. g. of order $v$ without commutative diagrams formed by two directed passes of length with the common starting and ending points. We introduce the upper bound for $E_R(d,v)$, which is one of the analogs of well known Even Circuit Theorem by P. Erdös’. The Erdös’ theorem establish the upper bound on maximal size of simple graphs without cycles of length $2n$. It is known to be sharp for the cases $n=2,3and5$ only. The situation with the upper bound for $E_d\left(v\right)$ is different: we prove that it is sharp for each $d\ge 2$. We introduce the girth of directed graph and establish tight upper and lower bounds on the order of directed cages, i.e. directed regular graphs of given girth and minimal order. The studies of regular directed graphs of large size (or small order) without small commutative diagrams, especially algebraic explicit constructions of them, are motivated by their applications to the design of turbo codes in Coding Theory and cryptographical algorithms. We introduce several new algebraic constructions of directed extremal graphs based on biregular generalized polygons, family of directed graphs of large girth with fixed degree.

The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry from a computational viewpoint. In this survey, we briefly describe some open problems of computational algebraic geometry which can be approached from such viewpoint. Some of the problems we discuss are the decomposition of Jacobians of genus two curves, automorphisms groups of algebraic curves and the corresponding loci in the moduli space of algebraic curves ${ℳ}_g$, inclusions among such loci, decomposition of Jacobians of algebraic curves with automorphisms, invariants of binary forms and the hyperelliptic moduli, theta functions of curves with automorphisms, etc. We decompose Jacobians of genus $3$ curves with automorphisms and determine the inclusions among the loci for algebraic curves with automorphisms of genus $3$ and $4$.