Let $K$ be a field of characteristic not two and $K(x,y,z)$ the rational function field over $K$ with three variables $x,y,z$. Let $G$ be a finite group of acting on $K(x,y,z)$ by monomial $K$-automorphisms. We consider the rationality problem of the fixed field $K(x,y,z)^G$ under the action of $G$, namely whether $K(x,y,z)^G$ is rational (that is, purely transcendental) over $K$ or not. We may assume that $G$ is a subgroup of $\mathrm{GL}(3,\mathbb{Z}) and the problem is determined up to conjugacy in $\mathrm{GL}(3,\mathbb{Z})$. There are 73 conjugacy classes of $G$ in $\mathrm{GL}(3,\mathbb{Z})$. By results of Endo-Miyata, Voskresenski\uı, Lenstra, Saltman, Hajja, Kang and Yamasaki, 8 conjugacy classes of 2-groups in $\mathrm{GL}(3,\mathbb{Z})$ have negative answers to the problem under certain monomial actions over some base field $K$, and the necessary and sufficient condition for the rationality of $K(x,y,z)^G$ over $K$ is given. In this paper, we show that the fixed field $K(x,y,z)^G$ under monomial action of $G$ is rational over $K$ except for possibly negative 8 cases of 2-groups and unknown one case of the alternating group of degree four. Moreover we give explicit transcendental bases of the fixed fields over $K$. For unknown case, we obtain an affirmative solution to the problem under some conditions. In particular, we show that if $K$ is quadratically closed field then $K(x,y,z)^G$ is rational over $K$. We also give an application of the result to 4-dimensional linear Noether's problem.
We give a short survey of the rationality problem for quasi-monomial actions which includes Noether's problem and the rationality problem for algebraic tori, and report some results on rationality problem in three recent papers Hoshi, Kang and Kitayama [HKKi], Hoshi, Kang and Kunyavskii [HKKu] and Hoshi and Yamasaki [HY].
Let $G$ be a finite 2-group and $K$ be a field satisfying that (i) $\fn{char}K\ne 2$, and (ii) $\sqrt{a}\in K$ for any $a\in K$. If $G$ acts on the rational function field $K(x,y,z)$ by monomial $K$-automorphisms, then the fixed field $K(x,y,z)^G$ is rational (= purely transcendental) over $K$. Applications of this theorem will be given.
Let $G$ be a finite subgroup of $\mathrm{Aut}_k(K(x_1, \ldots, x_n))$ where $K/k$ is a finite field extension and $K(x_1,\ldots,x_n)$ is the rational function field with $n$ variables over $K$. The action of $G$ on $K(x_1, \ldots, x_n)$ is called quasi-monomial if it satisfies the following three conditions (i) $σ(K)\subset K$ for any $σ\in G$; (ii) $K^G=k$ where $K^G$ is the fixed field under the action of $G$; (iii) for any $σ\in G$ and $1 \leq j \leq n$, $σ(x_j)=c_j(σ)\prod_{i=1}^n x_i^{a_{ij}}$ where $c_j(σ)\in K^\times$ and $[a_{i,j}]_{1\le i,j \le n} \in GL_n(\mathbb{Z})$. A quasi-monomi…
Let G be the (special) affine group, semidirect product of SL_n and C^n. In this paper we study the representation theory of G and in particular the question of rationality for V/G where V is a generically free G-representation. We show that the answer to this question is positive if the dimension of V is sufficiently large and V is indecomposable. We have a more precise theorem if V is a two-step extension 0 -> S -> V -> Q -> 0 with S, Q completely reducible.
Let $\mathfrak{Var}_k^G$ denote the category of pairs $(X,σ)$, where $X$ is a variety over $k$ and $σ$ is a group action on $X$. We define the Grothendieck ring for varieties with group actions as the free abelian group of isomorphism classes in the category $\mathfrak{Var}_k^G$ modulo a cutting and pasting relation. The multiplication in this ring is defined by the fiber product of varieties. This allows for motivic zeta-functions for varieties with group actions to be defined. This is a formal power series $\sum_{n=0}^{\infty}[\text{Sym}^n (X,σ)]t^n$ with coefficients in the Grothendieck rin…
We complete the study of rationality problem for hypersurfaces $X_t\subset \mathbb{P}^4$ of degree $4$ invariant under the action of the symmetric group $S_6$.
We introduce the notion of monomial group action and study some of its consequences for Groebner basis theory. As an application we prove a conjecture of V. Batyrev and O. Popov describing the Cox rings of Del Pezzo surfaces (of degree at least 3) as quotients of a polynomial ring by an ideal generated by quadrics.
Rationality problems of algebraic k-tori are closely related to rationality problems of the invariant field, also known as Noether's Problem. We describe how a function field of algebraic k-tori can be identified as an invariant field under a group action and that a k-tori is rational if and only if its function field is rational over k. We also introduce character group of k-tori and numerical approach to determine rationality of k-tori.
Sets of monomials separating Zariski closed orbits under diagonalizable group actions are characterized in terms of the monoid of zero-sum sequences over the character group. This is applied to compare the degree bounds for separating invariants and generating invariants of diagonalizable group actions.
For a hyperelliptic curve of genus $g$, a divisor in general position of degree $g+1$ is given by polynomial equations. There is an action from an algebraic group on the representations of divisors by polynomials which fixes divisor classes. This structure reduces the question of rationality of divisor classes to rationality of polynomials which is more easy to control.
We introduce equivariant versions of uniform rationality: given an algebraic group G, a G-variety is called G-uniformly rational (resp. G-linearly uniformly rational) if every point has a G-invariant open neighborhood equivariantly isomorphic to a G-invariant open subset of the affine space endowed with a G-action (resp. linear G-action). We establish a criterion for Gm-uniform rationality of affine variety equipped with hyperbolic Gm-action with a unique fixed point, formulated in term of their Altmann-Hausen presentation. We prove the Gm-uniform rationality of Koras-Russel threefolds of the …
Consider the diagonal action of $SL_n(K)$ on the affine space $X=V^{\oplus m}\oplus (V^*)^{\oplus q}$ where $V=K^n, K$ an algebraically closed field of arbitrary characteristic and $m,q>n$. We construct a "standard monomial" basis for the ring of invariants $K[X]^{SL_n(K)}$. As a consequence, we deduce that $K[X]^{SL_n(K)}$ is Cohen-Macaulay. We also present the first and second fundamental theorems for $SL_n(K)$-actions.
Consider the diagonal action of the special orthogonal group on the direct sum of a finite number of copies of the standard representation--the underlying field is assumed to be algebraically closed and of characteristic not equal to two. We construct a "standard monomial" basis for the ring of polynomial invariants for this action. We then deduce, by a deformation argument, our main result that this ring of polynomial invariants is Cohen-Macaulay. We give three applications of this result: (1) the first and second fundamental theorems of invariant theory for the above action; (2) Cohen-Macaul…
Fix a finite group $G$. We seek to classify varieties with $G$-action equivariantly birational to a representation of $G$ on affine or projective space. Our focus is odd-dimensional smooth complete intersections of two quadrics, relating the equivariant rationality problem with analogous Diophantine questions over nonclosed fields. We explore how invariants -- both classical cohomological invariants and recent symbol constructions -- control rationality in some cases.
We present a dynamical proof of the well-known fact that the Neron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field k of transcendence degree 1 over an algebraically closed field K of characteristic 0. More generally, we investigate the mechanism for which the local canonical height for a rational function f defined over k can take irrational values (at points in a local completion of k), providing examples in all degrees greater than 1. Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps…
This is a survey on (lack of) stable rationality over arbitrary fields (including algebraically closed fields). Topics addressed include: Rationality and unirationality, R-equivalence on rational points, Chow groups of zero-cycles, Galois action on the Picard group, Brauer group, higher unramified cohomology, global differentials, specialisation method (via R-equivalence), geometrically rational surfaces, cubic hypersurfaces.
This is a survey on the ancient question : Let G be a reductive group over an algebraically closed field k and let V be a vector space over k with an almost free linear action of G on V. Let k(V) denote the field of rational functions on V. Is the subfield of G-invariants of k(V) purely transcendental over k ? For G connected, this is still an open question. After a discussion of general matters (various notions of rationality, various notions of quotients, the no-name lemma), we consider several specific groups G. We then discuss the unramified Brauer group of a function field and describe th…
The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is represented by a balanced graph. Our main result is an explicit construction of that graph. As a consequence, we obtain algorithms to effectively compute the multidegree and Chow polytope of an arbitrary projective monomial curve. This generalizes an earlier degree formula due to Ranestad. The combinatorics underlying our construction is rather delicate, and it is bas…
We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A short survey on the structure of the Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's question on the rationality of the irreducible components of Chow varieties.
We classify Galois actions on Picard lattices of del Pezzo surfaces of degrees 1,2, and 3 giving rise to minimal surfaces with no cohomological obstructions to stable rationality.
Let $G$ be a finite subgroup of $GL_4(\bm{Q})$. The group $G$ induces an action on $\bm{Q}(x_1,x_2,x_3,x_4)$, the rational function field of four variables over $\bm{Q}$. Theorem. The fixed subfield $\bm{Q}(x_1,x_2,x_3,x_4)^G:=\{f\in\bm{Q}(x_1,x_2,x_3,x_4):σ\cdot f=f$ for any $σ\in G\}$ is rational (i.e.\ purely transcendental) over $\bm{Q}$, except for two groups which are images of faithful representations of $C_8$ and $C_3\rtimes C_8$ into $GL_4(\bm{Q})$ (both fixed fields for these two exceptional cases are not rational over $\bm{Q}$). There are precisely 227 such groups in $GL_4(\bm{Q})$ …
The projective orthogonal and symplectic groups $PO_n(F)$ and $PSp_n(F)$ have a natural action on the $F$ vector space $V' = M_n(F) \oplus ... \oplus M_n(F)$. Here we assume $F$ is an infinite field of characteristic not 2. If we assume there is more than one summand in $V'$, then the invariant fields $F(V')^{PO_n}$ and $F(V')^{PSp_n}$ are natural objects. They are, for example, the centers of generic algebras with the appropriate kind of involution. This paper considers the rationality properties of these fields, in the case $1,2$ or 4 are the highest powers of 2 that divide $n$. We derive ra…
The Gröbner stratum of a monomial ideal $\id{j}$ is an affine variety that parametrizes the family of all ideals having $\id{j}$ as initial ideal (with respect to a fixed term ordering). The Gröbner strata can be equipped in a natural way of a structure of homogeneous variety and are in a close connection with Hilbert schemes of subvarieties in the projective space $\PP^n$. Using properties of the Gröbner strata we prove some sufficient conditions for the rationality of components of $\hilb_{p(z)}^n$. We show for instance that all the smooth, irreducible components in $\hilb_{p(z)}^n$ (or in i…
In this paper we consider various problems involving the action of a reductive group $G$ on an affine variety $V$. We prove some general rationality results about the $G$-orbits in $V$. In addition, we extend fundamental results of Kempf and Hesselink regarding optimal destabilizing parabolic subgroups of $G$ for such general $G$-actions. We apply our general rationality results to answer a question of Serre concerning the behaviour of his notion of $G$-complete reducibility under separable field extensions. Applications of our new optimality results also include a construction which allows us…
We prove a decomposition theorem for the equivariant K-theory of actions of affine group schemes G of finite type over a field on regular separated noetherian algebraic spaces, under the hypothesis that the actions have finite geometric stabilizers and satisfy a rationality condition together with a technical condition which holds e.g. for G abelian or smooth. We describe in an Appendix various complicial bi-Waldhausen categories (in Thomason's terminology) modelling the equivariant K-theory of regular noetherian separated algebraic spaces.
Chemical reaction networks are often used to model and understand biological processes such as cell signaling. Under the framework of chemical reaction network theory, a process is modeled with a directed graph and a choice of kinetics, which together give rise to a dynamical system. Under the assumption of mass action kinetics, the dynamical system is polynomial. In this paper, we consider the ideals generated by the these polynomials, which are called steady-state ideals. Steady-state ideals appear in multiple contexts within the chemical reaction network literature, however they have yet to…
