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T. Shaska

Publications and source records attributed to T. Shaska.

At least 19 recordsLinked to original sources
arXiv2026

Deciding superellipticity and computing the Weierstrass normal form

Let \( \mathcal{S}_{g,n} \subset \mathcal{M}_g \) be the locus of curves of genus \( g \geq 2 \) admitting a model \( y^n = h(x) \) with \( h \) separable; such curves $C$ have a cyclic group \( C_n \leq \operatorname{Aut}(C) \) of order \( n \) with \( C/C_n \cong \mathbb{P}^1 \). % We give an algorithm which, given an absolutely irreducible plane model \( F(x,y) = 0 \) of a curve \( C \) over a field \( k_0 \) of characteristic zero, decides for which \( n \) the curve lies in \( \mathcal{S}_{g,n} \) and returns a model \( y^n = h(x) \) together with the birational transformation to it.

math.AG↗
arXiv2026

The invariant ring of degree-four rational maps on the projective line

Let $k$ be an algebraically closed field of characteristic zero. Degree-four rational maps on $\mathbb{P}^1$, up to conjugation, correspond to pairs of binary forms $(F,G)\in V_5\oplus V_3$. The associated invariant ring is the joint invariant ring $\mathcal{R}_{5,3}=k[V_5\oplus V_3]^{\mathrm{SL}_2}$. We determine a minimal generating set for $\mathcal{R}_{5,3}$, consisting of fifty explicit joint transvectants of degrees at most $18$. As consequences we describe the null cone of $V_5\oplus V_3$, construct six explicit absolute invariants that generate the function field of the moduli space $\mathcal{M}_4^1$, and give an effective criterion for determining conjugacy of degree-four rational maps.

math.AG↗
arXiv2026

Degrees of Genus-Two Split-Jacobian Loci and Humbert-Form Reconstruction

Let $\mathcal{L}_n \subset \mathcal{M}_2$ be the locus of genus-two curves admitting a maximal degree-$n$ elliptic subcover, cut out in $\mathbb{P}(2,4,6,10)$ by an irreducible weighted-homogeneous polynomial $F_n \in \mathbb{Z}[J_2,J_4,J_6,J_{10}]$. Let $\nu(n)$ be the degree of $X_1(n) \to X(1)$, let $G_{n^2}$ be the Siegel modular form of level one with divisor the Humbert surface $H_{n^2}$, and let $k(H_{n^2})$ be its weight. We prove that the meromorphic Siegel modular form $F_n(\tau)$ obtained from $F_n$ has a pole of order exactly $\nu(n)$ along the product locus, that $\chi_{10}^{\nu(n)} F_n(\tau)$ is a constant multiple of $G_{n^2}$, and that $\deg_w F_n = k(H_{n^2}) - 10\nu(n)$ for every $n \geq 2$, even or odd. We determine the restriction of $G_{n^2}$ to the product locus as an explicit product of modular polynomials and, for $n \geq 3$, its leading Fourier-Jacobi coefficient as a product of theta functions over the torsion points of exact order $n$, and we characterize $G_{n^2}$, up to scalar, as the unique form of its weight vanishing on a single torsion divisor. These data convert the computation of $F_n$ from elimination into a linear problem of the size the formula prescribes, which we carry out for $n=5$.

math.AG↗
arXiv2026

Hierarchical Grading in Large Language Models

We introduce Graded Large Language Models (GLLMs), an algebraic framework that equips the representation space of a transformer with a grading and propagates the induced weighted scalar action through embeddings, self-attention, and the training objective. The construction extends the theory of graded neural networks and graded transformers to autoregressive language models while preserving expressive power, asymptotic computational complexity, and inference cost. The governing geometric picture is that of geometric invariant theory. The benefit of a grading is expressed by a Kempf--Ness functional on the grading torus; the grades that improve upon the uniform architecture form an open convex cone whose membership is decided by a Hilbert--Mumford-type criterion pairing a grade direction against two measurable profiles of the target and the data; the optimal grades are the coincidence point of two moment maps, given in closed form; and the ordinary transformer appears as a semistable isotropic point on the boundary of the cone: one member of a larger graded family rather than a distinguished optimum. Separately, for level-stratified targets we prove a minimax separation between the graded prior and its absence: over all estimators the risks of the graded and uniform target classes separate throughout an explicit window of sample sizes, by a factor that decays exponentially in the number of levels under geometric stratification. Both profiles are estimable offline, so the optimal grades solve a convex program certified before training begins. Because the grading is absorbed into the learned parameters after training, every GLLM compiles to a standard transformer of identical architecture and inference complexity.

cs.LG↗
arXiv2026

Graded Keller maps and the Jacobian Conjecture

Rational points on weighted projective spaces are sparse: a point of $\mathbb{P}^n(\mathbb{Q})$ lifts along the Veronese morphism only when a Kummer condition holds at every prime (arXiv:2509.02319). We ask what this viewpoint says about the Jacobian Conjecture. The map recently announced as a counterexample is equivariant for the grading $\mathrm{wt}(x,y,z)=(1,-1,-2)$, and we show that the sign pattern of such a grading decides everything. If the weights are all positive, the setting of weighted projective spaces, an equivariant Keller map is always an automorphism, so no counterexample can be graded that way. In dimension two the same holds for every sign pattern. The Keller condition itself descends to the quotient, where it says that the Jacobian of the quotient map vanishes to order two along the contracted locus. Over $\mathbb{Q}$ the image of the counterexample is a thin set. But along the line where the group action has stabilizer $\mu_2$, the two preimages on the contracted locus are rational exactly when $-a$ is a square, so the Kummer condition of the positive-weight theory reappears on the stacky stratum. We propose a classification of equivariant polynomial maps by the signature of the weights, with weighted projective geometry as the positive case.

math.AG↗
arXiv2026

A Unified Mathematical Framework for Distributed Data Fabrics: Categorical Hypergraph Models

Current distributed data fabrics lack a rigorous mathematical foundation, often relying on ad-hoc architectures that struggle with consistency, lineage, and scale. We propose a mathematical framework for data fabrics, unifying heterogeneous data management in distributed systems through a hypergraph-based structure \( \mathcal{F} = (D, M, G, T, P, A) \). Datasets, metadata, transformations, policies, and analytics are modeled over a distributed system \( \Sigma = (N, C) \), with multi-way relationships encoded in a hypergraph \( G = (V, E) \). A categorical approach, with datasets as objects and transformations as morphisms, supports operations like data integration and federated learning. The hypergraph is embedded into a modular tensor category, capturing relational symmetries via braided monoidal structures, with geometric analogies to Hurwitz spaces enriching the algebraic modeling. We prove the NP-hardness of critical tasks, such as schema matching and dynamic partitioning, and propose spectral methods and symmetry-based alignments for scalable solutions. The framework ensures consistency, completeness, and causality under CAP and CAL theorems, leveraging sparse incidence matrices and braiding actions for fault-tolerant operations.

cs.DB↗
arXiv2019

Rational points in the moduli space of genus two

We build a database of genus 2 curves defined over $\mathbb Q$ which contains all curves with minimal absolute height $h \leq 5$, all curves with moduli height $\mathfrak h \leq 20$, and all curves with extra automorphisms in standard form $y^2=f(x^2)$ defined over $\mathbb Q$ with height $h \leq 101$. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli space $\mathcal M_2$ for which the field of moduli is a field of definition is discussed and some open problems are presented.

math.AG↗
arXiv2018

On automorphisms of algebraic curves

An irreducible, algebraic curve $\mathcal X_g$ of genus $g\geq 2$ defined over an algebraically closed field $k$ of characteristic $\mbox{char } \, k = p \geq 0$, has finite automorphism group $\mbox{Aut} (\mathcal X_g)$. In this paper we describe methods of determining the list of groups $\mbox{Aut} (\mathcal X_g)$ for a fixed $g\geq 2$. Moreover, equations of the corresponding families of curves are given when possible.

math.AG↗
arXiv2015

Theta functions of superelliptic curves

In this short survey we give a description of the theta functions of algebraic curves, half-integer theta-nulls, and the fundamental theta functions. We describe how to determine such fundamental theta functions and describe the components of the moduli space in terms of such functions.

math.CV↗
arXiv2014

Heights on algebraic curves

In these lectures we cover basics of the theory of heights starting with the heights in the projective space, heights of polynomials, and heights of the algebraic curves. We define the minimal height of binary forms and moduli height for algebraic curves and prove that the moduli height of superelliptic curves $\mathcal H (f) \leq c_0 \tilde H (f)$ where $c_0$ is a constant and $\tilde H$ the minimal height of the corresponding binary form. For genus $g=2$ and 3 such constant is explicitly determined. Furthermore, complete lists of curves of genus 2 and genus 3 hyperelliptic curves with height 1 are computed.

math.NT↗
arXiv2014

Genus 3 hyperelliptic curves with (2, 4, 4)-split Jacobians

We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over $\mathbb C$. The family of genus 3 hyperelliptic curves which have a degree 2 cover to an elliptic curve $E$ and degree 4 covers to elliptic curves $E_1$ and $E_2$ is a 2-dimensional subvariety of the hyperelliptic moduli $\mathcal H_3$. We determine this subvariety explicitly. For any given moduli point $\mathfrak p \in \mathcal H_3$ we determine explicitly if the corresponding genus 3 curve $\mathcal X$ belongs or not to such family. When it does, we can determine elliptic subcovers $E$, $E_1$, and $E_2$ in terms of the absolute invariants $t_1, \dots, t_6$ as in \cite{hyp_mod_3}. This variety provides a new family of hyperelliptic curves of genus 3 for which the Jacobians completely split. The sublocus of such family when $E_1$ is isomorphic to $E_2$ is a 1-dimensional variety which we determine explicitly. We can also determine $\mathcal X$ and $E$ starting form the $j$-invariant of $E_1$.

math.AG↗
arXiv2013

Quantum codes from superelliptic curves

Let $\X$ be an algebraic curve of genus $g \geq 2$ defined over a field $\F_q$ of characteristic $p > 0$. From $\X$, under certain conditions, we can construct an algebraic geometry code $C$. If the code $C$ is self-orthogonal under the symplectic product then we can construct a quantum code $Q$, called a QAG-code. In this paper we study the construction of such codes from curves with automorphisms and the relation between the automorphism group of the curve $\X$ and the codes $C$ and $Q$.

cs.IT↗
arXiv2013

The 2-Weierstrass points of genus 3 hyperelliptic curves with extra automorphisms

For each group $G$, $(|G| > 2)$ \, which acts as a full automorphism group on a genus 3 hyperelliptic curve, we determine the family of curves which have 2-Weierstrass points. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. The 1-dimensional families that we discover have the property that they contain only genus 0 components.

math.AG↗
arXiv2013

Bielliptic curves of genus 3 in the hyperelliptic moduli

In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field $k$ and the intersection of the moduli space $\M_3^b$ of such curves with the hyperelliptic moduli $\H_3$. Such intersection $§$ is an irreducible, 3-dimensional, rational algebraic variety. We determine the equation of this space in terms of the $Gl(2, k)$-invariants of binary octavics as defined in \cite{hyp_mod_3} and find a birational parametrization of $§$. We also compute all possible subloci of curves for all possible automorphism group $G$. Moreover, for every rational moduli point $\p \in §$, such that $| \Aut (\p) | > 4$, we give explicitly a rational model of the corresponding curve over its field of moduli in terms of the $Gl(2, k)$-invariants.

math.AG↗
arXiv2013

On the automorphism groups of some AG-codes based on $C_{a, b}$ curves

We study $C_{a, b}$ curves and their applications to coding theory. Recently, Joyner and Ksir have suggested a decoding algorithm based on the automorphisms of the code. We show how $C_{a, b}$ curves can be used to construct MDS codes and focus on some $C_{a, b}$ curves with extra automorphisms, namely $y^3=x^4+1, y^3=x^4-x, y^3-y=x^4$. The automorphism groups of such codes are determined in most characteristics.

cs.IT↗
arXiv2013

Subvarieties of the hyperelliptic moduli determined by group actions

Let $\mathcal H_g$ be the moduli space of genus $g$ hyperelliptic curves. In this note, we study the locus $\mathcal L$ in $\mathcal H_g$ of curves admitting a $G$-action of given ramification type $σ$ and inclusions between such loci. For each genus we determine the list of all possible groups, the inclusions among the loci, and the corresponding equations of the generic curve in $\mathcal L$. The proof of the results is based solely on representations of finite subgroups of $PGL_2 (\mathbb C)$ and the Riemann-Hurwitz formula.

math.AG↗