arXiv subjects

Michael Chitayat

Publications and source records attributed to Michael Chitayat.

7 recordsLinked to original sources

arXiv2026

Locally finite sets of derivations

Given an algebra B over a field k, we study conditions under which a Lie subalgebra of Der(B) is locally finite as a set of derivations. As an application of our results, we show that if X is a quasi-affine variety over an arbitrary field k, and if L is a finitely generated solvable Lie subalgebra of Der O(X) consisting of locally finite derivations, then L is locally finite. If, moreover, k is algebraically closed and of characteristic zero, and X is irreducible and affine, then L is integrable.

math.AC↗

arXiv2026

The Isomorphism Classes of the Surfaces $x_1^{a_1} + x_2^{a_2} + x_3^{a_3} + 1 = 0$

Let $f = x_1^{a_1} + x_2^{a_2} + x_3^{a_3} + 1 \in \mathbb{C}[x_1,x_2,x_3]$ and let $g = y_1^{b_1} + y_2^{b_2} + y_3^{b_3} + 1 \in \mathbb{C}[y_1,y_2,y_3]$ where $a_1,a_2,a_3,b_1,b_2,b_3 \geq 2$. We prove that the surfaces $V(f) \subset \mathbb{A}^3$ and $V(g) \subset \mathbb{A}^3$ are isomorphic if and only if $(a_1,a_2,a_3) = (b_1,b_2,b_3)$ up to a permutation of the entries.

math.AG↗

arXiv2025

The Fundamental Group of a Compact Riemann Surface via Branched Covers

Let $X$ be a compact Riemann surface of genus $g$ and let $x \in X$. We derive the classical presentation of $\pi_1(X,x)$ (i.e the one given by $2g$ generators $a_1,b_1, \dots, a_g,b_g$ and the relation $\prod_{i=1}^g[a_i,b_i] = 1$) from the description of $X$ as a branched cover $f : X \to \mathbb{C}\mathbb{P}^1$.

math.AT↗

arXiv2023

The Rigid Pham-Brieskorn Threefolds

We show that a $3$-dimensional Pham-Brieskorn hypersurface $\{ X_0^{a_0} + X_1^{a_1} + X_2^{a_2} + X_3^{a_3}=0\}$ in $\mathbb{A}^4$ such that $\min\{a_0, a_1, a_2, a_3 \} \geq 2$ and at most one element $i$ of $\{0,1,2,3\}$ satisfies $a_i = 2$ does not admit a non-trivial action of the additive group $\mathbb{G}_a$.

math.AG↗

arXiv2023

Rationality of weighted hypersurfaces of special degree

Let $X \subset \mathbb{P}(w_0, w_1, w_2, w_3)$ be a quasismooth well-formed weighted projective hypersurface and let $L = lcm(w_0,w_1,w_2,w_3)$. We characterize when $X$ is rational under the assumption that $L$ divides $deg(X)$ by combining an algebraic proof of rationality valid in all dimensions with a new result on numerical semigroups. As applications, we give new examples of families of normal projective rational varieties with quotient singularities and ample canonical divisor; we also determine precisely which affine Pham-Brieskorn threefolds are rational.

math.AG↗

arXiv2021

Locally nilpotent derivations of graded integral domains and cylindricity

Let B be a commutative $\mathbb{Z}$-graded domain of characteristic zero. An element f of B is said to be cylindrical if it is nonzero, homogeneous of nonzero degree, and such that $B_{(f)}$ is a polynomial ring in one variable over a subring. We study the relation between the existence of a cylindrical element of B and the existence of a nonzero locally nilpotent derivation of B. Also, given d > 0, we give sufficient conditions that guarantee that every derivation of $B^{(d)} = \oplus_i B_{di}$ can be extended to a derivation of B. We generalize some results of Kishimoto, Prokhorov and Zaidenberg that relate the cylindricity of a polarized projective variety (Y,H) to the existence of a nontrivial G_a-action on the affine cone over (Y,H).

math.AG↗

arXiv2019

On the rigidity of certain Pham-Brieskorn rings

Fix a field $k$ of characteristic zero. If $a_1, ..., a_n$ ($n>2$) are positive integers, the integral domain $B = k[X_1, ..., X_n] / ( X_1^{a_1} + ... + X_n^{a_n} )$ is called a Pham-Brieskorn ring. It is conjectured that if $a_i > 1$ for all $i$ and $a_i=2$ for at most one $i$, then $B$ is rigid. (A ring $B$ is said to be rigid if the only locally nilpotent derivation $D: B \to B$ is the zero derivation.) We give partial results towards the conjecture.

math.AC↗