Narrow Results

An induced additive action on a projective variety $X\subseteq {\mathbb{P}}^n$ is a regular action of the group ${𝔾}_a^m$ on $X$ with an open orbit, which can be extended to a regular action on the ambient projective space ${\mathbb{P}}^n$. In this work, we classify all projective hypersurfaces admitting an induced additive action with a finite number of orbits.

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.

Fix a group G and let X be an algebraic variety over an algebraically closed field k of characteristic zero. We investigate the invertibility of algebraic cellular automata, namely, G-equivariant uniformly continuous self-maps whose local defining maps are induced by morphisms of algebraic varieties $X^M\to X$ where $M\subset G$ is a finite memory set. When G is locally embeddable into finite groups (LEF), we show that the inverses of reversible algebraic cellular automata are automatically algebraic cellular automata and thus computable in polynomial time. Generalizations are also obtained for finite product Hopfian pointed object alphabets in concrete categories. Moreover, we prove that for algebraic cellular automata, the notions of reversibility and injectivity are equivalent whenever G is surjunctive and the field k is, additionally, uncountable of arbitrary characteristic.

In our earlier work on a new functor from $E_6$-Mod to $E_7$-Mod, we found a one-parameter ($c$) family of inhomogeneous first-order differential operator representations of the simple Lie algebra of type $E_7$ in $27$ variables. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector $\overrightarrow{a}\in {ℂ}^{27}\backslash \{\overrightarrow{0}\}$, we prove that the space forms an irreducible $E_7$-module for any $c\in ℂ$ if $\overrightarrow{a}$ is not on an explicitly given projective algebraic variety. Certain equivalent combinatorial properties of the basic oscillator representation of $E_6$ over its 27-dimensional module play key roles in our proof. Our result can also be used to study free bosonic field irreducible representations of the corresponding affine Kac–Moody algebra.

In this paper, we define the geometric and algebraic tangent cones at infinity of algebraic varieties and establish the following version at infinity of Whitney’s theorem [Local properties of analytic varieties, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (Princeton University Press, Princeton, N. J., 1965), pp. 205–244; Tangents to an analytic variety, Ann. of Math.81 (1965) 496–549]: The geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of the geometric tangent cone at infinity using the global Łojasiewicz inequality with explicit exponents for complex algebraic varieties. Moreover, we show that the tangent cone at infinity of a complex algebraic variety is actually the part at infinity of this variety [G.-M. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra, 2nd extended edn. (Springer, Berlin, 2008)]. We also show that the tangent cone at infinity of a complex algebraic variety can be computed using Gröbner bases.

Let ${𝒢}=\{g_1,\dots ,g_n\}$ be a set of elements in the polynomial ring $R=ℂ[z_1,\dots ,z_N]$, let $I\subset R$ denote the ideal generated by the elements of ${𝒢}$, and let $\sqrt{I}$ denote the radical of $I$. There is a unique decomposition $\sqrt{I}=P_1\cap \cdots \cap P_k$ with each $P_i$ a prime ideal corresponding to a minimal associated prime of $I$ over $R$. Let $V({𝒢})=V(I)$ denote the reduced algebraic set corresponding to the common zeroes of the elements of ${𝒢}$. Techniques from numerical algebraic geometry can be used to determine the numerical irreducible decomposition of $V({𝒢})$ over $ℂ$. This corresponds to producing a witness set for $V(P_i)$ for each $i=1,\dots ,k$ together with the degree and dimension of $V(P_i)$ (a point in a witness set for $V(P_i)$ can be considered as a numerical approximation for a general point on $V(P_i)$). The purpose of this paper is to show how to extend these results taking into account the field of definition for the polynomial system. In particular, let $F$ be a number field (i.e. a finite field extension of $\mathbb{Q}$) and let ${𝒢}=\{g_1,\dots ,g_n\}$ be a set of elements in $S=F[z_1,\dots ,z_N]$. We show how to extend techniques from numerical algebraic geometry to determine the numerical irreducible decomposition of $V({𝒢})$ over $F$.

Three different notions of geometric degeneration for finite dimensional algebras are compared. It is proved that for algebras A₀ and A₁ of same dimension, if A₀ is a CB-degeneration of A₁, then A₀ is a degeneration of A₁ in the classical sense; moreover, if additionally A₀ and A₁ are basic, then A₀ is a CB-degeneration of A₁ if and only if A₀ is a rigid degeneration of A₁. It is also shown that CB-degenerations along affine lines in the different dimension case do not differ so much from those in the equal dimension case.

We show that the relation ≤_cb, induced from the CB-degeneration relation on the disjoint union ∐_d∈ℕ(alg_d(k)/≅), is a partial order, where for d ∈ ℕ the set alg_d(k)/≅ consists of the isomorphism classes of k-algebra structures on the space k^d.