Related papers: Euler-symmetric projective toric varieties and add…
Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{G}_a$ be the additive group of $\mathbb{K}$. We say that an irreducible algebraic variety $X$ of dimension $n$ over the field $\mathbb{K}$ admits an…
By an additive action on an algebraic variety $X$ of dimension $n$ we mean a regular action $\mathbb{G}_a^n \times X \to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. We prove that if a complete toric variety…
Euler-symmetric projective varieties are nondegenerate projective varieties admitting many C*-actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. We show that…
An induced additive action on a projective variety $X\subseteq\mathbb{P}^n$ is a regular action of the group $\mathbb{G}_a^n$ on $X$ with an open orbit that can be extended to a regular action on $\mathbb{P}^n$. Such actions are known to…
A $\mathbb{C}^{*}$-action on a projective variety $X$ is said to be of Euler type at a nonsingular fixed point $x$ if the isotropy action of $\mathbb{C}^{*}$ on $T_{x}X$ is by scalar multiplication. In this paper, it's proven that a smooth…
In this paper, we study complete simplicial toric varieties admitting faithful actions of large symmetric groups. First, we correct a recent classification result by Esser, Ji, and Moraga concerning $4$-dimensional toric varieties with…
Euler-symmetric projective varieties, introduced by Baohua Fu and Jun-Muk Hwang in 2020, are nondegenerate projective varieties admitting many $\mathbb{C}^{\times}$-actions of Euler type. They are quasi-homogeneous and uniquely determined…
By an additive action on an algebraic variety $X$ we mean a regular effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. In this paper, we give a classification of additive…
By an additive action on an algebraic variety $X$ we mean a regular effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. In this paper, we give a uniqueness criterion for…
Let X be an algebraic projective variety in {\bf P}^n. Denote by {\cal C}_{\lambda} the space of all effective cycles on X whose homology class is \lambda \in H_{2p} (X,{\bf Z}). It is easy to show that {\cal C}_{\lambda} is an algebraic…
An additive action on an algebraic variety is an effective action of the vector group with an open orbit. We describe projective surfaces with du Val singularities that admit an additive action with a finite number of orbits. In particular,…
This is a continuation of arXiv: 2408.03012. We answer affirmatively Question 5.10 posed in the previous article. More precisely, let $(X, \omega)$ be a conical symplectic variety of dimension $2n$ with $wt(\omega) = 2$, which has a…
Let X be a normal variety endowed with an algebraic torus action. An additive group action $\alpha$ on X is called vertical if a general orbit of $\alpha$ is contained in the closure of an orbit of the torus action and the image of the…
We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler…
An induced additive action on a projective variety $X \subseteq \mathbb{P}^n$ is a regular action of the group $\mathbb{G}_a^m$ on $X$ with an open orbit, which can be extended to a regular action on the ambient projective space…
The motivation for this paper is to detect when an irreducible projective variety V is not toric. We do this by analyzing a Lie group and a Lie algebra associated to V. If the dimension of V is strictly less than the dimension of the above…
Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T stands for the split algebraic torus. Our main result is a classification of additive group actions on X that are normalized by the T-action. This…
We introduce the AdditiveToricVarieties package for Macaulay2, a software system for algebraic geometry and commutative algebra, with methods for working with additive group actions on complete toric varieties. More precisely, we implement…
We describe complete simplicial toric varieties on which a unipotent group acts with a finite number of orbits. We also provide a complete list of such varieties in the case where the dimension is equal to 2.
We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We…