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Daniel Daigle

Publications and source records attributed to Daniel Daigle.

At least 19 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↗
arXiv2024

Decorated trees

We study a class of combinatorial objects that we call "decorated trees". These consist of vertices, arrows and edges, where each edge is decorated by two integers (one near each of its endpoints), each arrow is decorated by an integer, and the decorations are required to satisfy certain conditions. The class of decorated trees includes different types of trees used in algebraic geometry, such as the Eisenbud and Neumann diagrams for links of singularities and the Neumann diagrams for links at infinity of algebraic plane curves. By purely combinatorial means, we recover some formulas that were previously understood to be "topological". In this way, we extend the generality of those formulas and show that they are in fact "combinatorial".

math.AG↗
arXiv2024

Automorphisms of the ring of invariants of the binary quintic representation of SL2

Let k^[6] denote a polynomial ring in 6 variables over an algebraically closed field k of characteristic zero and consider the action of SL2(k) on k^[6] induced by the irreducible representation of SL2 of degree 5 (the binary quintic representation). We consider the ring Q = (k^[6])^SL2 of invariant polynomials and show that Aut_k(Q) = u(k), the unit group of k, where Aut_k(Q) is the group of k-algebra automorphisms of Q. Based on this result, we show that the group of SL2-equivariant polynomial automorphisms of k^[6] is isomorphic to u(k).

math.AC↗
arXiv2023

Rigidity of Graded Integral Domains and of their Veronese Subrings

A ring R is said to be rigid if the only locally nilpotent derivation of R is the zero derivation. Let G be an abelian group, and B = (direct sum of B_i for i in G) be a G-graded commutative integral domain of characteristic 0. For each subgroup H of G, consider the Veronese subring B(H) of B, defined by B(H) = (direct sum of the B_i for i in H). We study the following questions. If B is non-rigid, does it follow that B(H) is non-rigid? Can derivations of B(H) be extended to derivations of B? What are the properties of the set of subgroups H of G such that B(H) is non-rigid?

math.AC↗
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↗
arXiv2020

The freeness property for locally nilpotent derivations of k[x,y,z]

We prove Freudenburg's Freeness Conjecture: Let B be the polynomial ring in three variables over a field of characteristic zero, let D : B --> B be a nonzero locally nilpotent derivation, and let A = ker(D). Then B is a free A-module, and there exists a basis $(e_i)_{i \in \mathbb{N}}$ of B such that deg$_D(e_i) = i$ for all $i \in \mathbb{N}$.

math.AC↗
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↗
arXiv2019

Field generators in two variables and birational endomorphisms of $\mathbb{A}^2$

This article is a survey of two subjects: the first part is devoted to field generators in two variables, and the second to birational endomorphisms of the affine plane. Each one of these subjects originated in Abhyankar's seminar in Purdue University in the 1970s. Note that the part on field generators is more than a survey, since it contains a considerable amount of new material.

math.AG↗
arXiv2019

Locally nilpotent sets of derivations

Let B be an algebra over a field k and let Der(B) be the set of k-derivations from B to B. We define what it means for a subset of Der(B) to be a locally nilpotent set. We prove some basic results about that notion and explore the following questions. Let L be a Lie subalgebra of Der(B); if every element of L is a locally nilpotent derivation then does it follow that L is a locally nilpotent set? Does it follow that L is a nilpotent Lie algebra?

math.AC↗
arXiv2018

Structure of the Newton tree at infinity of a polynomial in two variables

Let $f:\mathbb{C}^2 \to \mathbb{C}$ be a polynomial map. Let $\mathbb{C}^2 \subset X$ be a compactification of $\mathbb{C}^2$ where $X$ is a smooth rational compact surface and such that there exists a morphism of varieties $\Phi :X\to \mathbb{P}^1$ which extends $f$. Put $\mathcal{D}=X\setminus \mathbb{C}^2$; $\mathcal{D}$ is a curve whose irreducible components are smooth rational compact curves and all its singularities are ordinary double points. The dual graph of $\mathcal{D}$ is a tree. We are interested in this tree, and we analyse its complexity in terms of the genus of the generic fiber of $f$.

math.AG↗
arXiv2018

Rings with trivial FML-invariant

Let $k$ be a field of characteristic zero and $B$ a commutative integral domain that is also a finitely generated $k$-algebra. It is well known that if $k$ is algebraically closed and the "Field Makar-Limanov" invariant FML$(B)$ is equal to $k$, then $B$ is unirational over $k$. This article shows that, when $k$ is not assumed to be algebraically closed, the condition FML$(B)=k$ implies that there exists a nonempty Zariski-open subset $U$ of Spec$(B)$ with the following property: for each prime ideal $\mathfrak{p} \in U$, the $\kappa(\mathfrak{p})$-algebra $\kappa(\mathfrak{p}) \otimes_k B$ can be embedded in a polynomial ring in $n$ variables over $\kappa(\mathfrak{p})$, where $n=\dim B$ and $\kappa(\mathfrak{p}) = B_{\mathfrak{p}}/{\mathfrak{p}}B_{\mathfrak{p}}$.

math.AG↗
arXiv2013

Very good and very bad field generators

Let k be a field. A "field generator" is a polynomial F in k[X,Y] satisfying k(F,G) = k(X,Y) for some G in k(X,Y). If G can be chosen in k[X,Y], we call F a "good field generator"; otherwise, F is a "bad field generator". These notions were first studied by Abhyankar, Jan and Russell in the 1970s. The present paper introduces and studies the notions of "very good" and "very bad" field generators. We give theoretical results as well as new examples of bad and very bad field generators.

math.AG↗
arXiv2013

Compositions of birational endomorphisms of the affine plane

Let A^2 denote the affine plane over an algebraically closed field of arbitrary characteristic. Besides contributing several new results in the general theory of birational endomorphisms of A^2, this article describes certain classes of birational endomorphisms f defined by requiring that the missing curves or contracting curves of f are lines. The last part of the article is concerned with the monoid structure of the set of birational endomorphisms of A^2.

math.AG↗
arXiv2013

Generally rational polynomials in two variables

Let k be an algebraically closed field. A polynomial F in k[X,Y] is said to be "generally rational" if, for almost all c in k, the curve " F= c '' is rational. It is well known that, if char(k)=0, F is generally rational iff there exists G in k(X,Y) such that k(F,G)=k(X,Y). We give analogous results valid in arbitrary characteristic.

math.AG↗
arXiv2012

Linear systems associated to unicuspidal rational plane curves

A curve C in the projective plane is called non-negative if the self-intersection number of C after the minimal resolution of singularities of C is non-negative. Given a unicuspidal rational plane curve C with singular point P, we study the unique pencil Lambda_C on the projective plane satisfying C is in Lambda_C and P is its unique base point. We show that the general member of Lambda_C is a rational curve if and only if the curve C is non-negative. We also show that in such a case then Lambda_C has a dicritical of degree 1. Note that all currently known unicuspidal rational curves C in the projective plane are non-negative.

math.AG↗