Arbitrary Field Of Characteristic Zero Research Articles

Rational local unitary invariants of symmetrically mixed states of two qubits

We compute the field of rational local unitary invariants for locally maximally mixed states and symmetrically mixed states of two qubits. In both cases, we prove that the field of rational invariants is purely transcendental. We also construct explicit geometric quotients and prove that they are always rational. All the results are obtained by working over the field of real numbers, employing methods from classical and geometric invariant theory over arbitrary fields of characteristic zero.

On irreducible symplectic varieties of K3[n]-type in positive characteristic

We show that there is a good notion of irreducible symplectic varieties of K3[n]-type over an arbitrary field of characteristic zero or p>n+1. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus' crystalline Torelli theorem for supersingular K3 surfaces. For applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.

Twisted forms of commutative monoid structures on affine spaces

In this paper we study commutative algebraic monoid structures on affine spaces over an arbitrary field of characteristic zero. We obtain full classification of such structures on AK2 and AK3 and describe some generalizations on AKn for any dimension n.

Double Points of Free Projective Line Arrangements

Abstract We prove the Anzis–Tohăneanu conjecture, that is, the Dirac–Motzkin conjecture for supersolvable line arrangements in the projective plane over an arbitrary field of characteristic zero. Moreover, we show that a divisionally free arrangements of lines contain at least one double point that can be regarded as the Sylvester–Gallai theorem for some free arrangements. This is a corollary of a general result that if you add a line to a free projective line arrangement, then that line has to contain at least one double point. Also, we prove some conjectures and one open problems related to supersolvable line arrangements and the number of double points.

Isomorphism theorems for Galois groups of splitting fields of polynomials*

Let K be an arbitrary field of characteristic zero with an absolute value on it. We show that if F and G are monic and normal polynomials of of the same degree with coefficients close enough to each other with respect to this absolute value, then the Galois groups of the splitting fields of F and G over K are isomorphic. We point out a quantitative result and discuss some special cases and related problems.

Finite generation of extensions of associated graded rings along a valuation

In this paper we consider the question of when the associated graded ring along a valuation, gr ν ∗ ( S ) , is a finite gr ν ∗ ( R ) -module, where S is a normal local ring which lies over a normal local ring R and ν ∗ is a valuation of the quotient field of S which dominates S. We begin by discussing some examples and results, allowing us to refine the conditions under which finite generation can hold. We must impose the condition that the extension of valuations is defectless and performs a birational extension of R along the valuation to obtain finite generation (replacing S with the local ring of the quotient field of S determined by the valuation which lies over the extension of R). With these assumptions, we have that finite generation holds, when R is a two-dimensional excellent local ring. Our main result (in Theorem 1.5) is to show that for an arbitrary valuation in an algebraic function field over an arbitrary field of characteristic zero, after a birational extension along the valuation, we always have finite generation (all finite extensions of valued fields are defectless in characteristic zero). This generalizes an earlier result, in [Cutkosky, Math. Proc. Cambridge Soc. 106 (2016) 233–255], showing that finite generation holds (after a birational extension) with the additional assumptions that ν has rank 1 and has an algebraically closed residue field. There are essential difficulties in removing these assumptions, which are addressed in the proof in this paper. We obtain general results for unramified extensions of excellent local rings in Proposition 1.7, showing that after blowing up, the extension of associated graded rings is finitely generated of an extremely simple form. This proposition plays an essential role in the Proof of Theorem 1.5.

Cohomologie non ramifiée de degré 3 : variétés cellulaires et surfaces de del Pezzo de degré au moins 5

We consider geometrically cellular varieties $X$ over an arbitrary field of characteristic zero. We study the quotient of the third unramified cohomology group $H^3_{nr}(X,\mathbb{Q}/\mathbb{Z}(2))$ by its constant part. For $X$ a smooth compactification of a universal torsor over a geometrically rational surface, we show that this quotient if finite. For $X$ a del Pezzo surface of degree $\geq 5$, we show that this quotient is zero, unless $X$ is a del Pezzo surface of degree 8 of a special type.

QUOTIENTS OF CONIC BUNDLES

Let k be an arbitrary field of characteristic zero. In this paper we study quotients of k-rational conic bundles over by finite groups of automorphisms. We construct smooth minimal models for such quotients. We show that any quotient is birationally equivalent to a quotient of other k-rational conic bundle cyclic group $$ {\mathrm{\mathfrak{C}}}_{2^k} $$ of order 2 k , dihedral group $$ {\mathfrak{D}}_{2^k} $$ of order 2 k , alternating group $$ {\mathfrak{A}}_4 $$ of degree 4, symmetric group $$ {\mathfrak{S}}_4 $$ of degree 4 or alternating group $$ {\mathfrak{A}}_5 $$ of degree 5 effectively acting on the base of the conic bundle. Also we construct infinitely many examples of such quotients which are not k-birationally equivalent to each other.

Polynomials with constant Hessian determinants in dimension three

In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension n≤3 over a field K of characteristic zero. We do this by extending the following result for n≤2 by F. Dillen to n≤3: If f is a polynomial of degree larger than 2 in n≤3 variables such that the Hessian determinant of f is constant, then after a suitable linear transformation (replacing f by f(Tx) for some T∈GLn(K)), the Hessian matrix of f becomes zero below the anti-diagonal. The result does not hold for larger n.The proof of the case det⁡Hf∈K⁎ is based on the following result, which in turn is based on the already known case det⁡Hf=0: If f is a polynomial in n≤3 variables such that det⁡Hf≠0, then after a suitable linear transformation, there exists a positive weight function w on the variables such that the Hessian determinant of the w-leading part of f is nonzero. This result does not hold for larger n either (even if we replace ‘positive’ by ‘nontrivial’ above).In the last section, we show that the Jacobian conjecture holds for gradient maps over the reals whose linear part is the identity map, by proving that such gradient maps are translations (i.e. have degree 1) if they satisfy the Keller condition. We do this by showing that this problem is the polynomial case of the main result of [13]. For polynomials in dimension n≤3, we generalize this result to arbitrary fields of characteristic zero.

Existence of standard models of conic fibrations over non-algebraically-closed fields

We prove an analogue of Sarkisov's theorem on the existence of a standard model of a conic fibration over an algebraically closed field of characteristic different from two for three-dimensional conic fibrations over an arbitrary field of characteristic zero with an action of a finite group. Bibliography: 16 titles.

Rationality of the quotient of ℙ2 by finite group of automorphisms over arbitrary field of characteristic zero

Abstract Let $$\Bbbk$$ be a field of characteristic zero and G be a finite group of automorphisms of projective plane over $$\Bbbk$$. Castelnuovo’s criterion implies that the quotient of projective plane by G is rational if the field $$\Bbbk$$ is algebraically closed. In this paper we prove that $${{\mathbb{P}_\Bbbk ^2 } \mathord{\left/ {\vphantom {{\mathbb{P}_\Bbbk ^2 } G}} \right. \kern-\nulldelimiterspace} G}$$ is rational for an arbitrary field $$\Bbbk$$ of characteristic zero.

Nearby motives and motivic nearby cycles

We prove that the construction of motivic nearby cycles, introduced by Jan Denef and Francois Loeser, is compatible with the formalism of nearby motives, developed by Joseph Ayoub. Let $$k$$ be an arbitrary field of characteristic zero, and let $$X$$ be a smooth quasi-projective $$k$$ -scheme. Precisely, we show that, in the Grothendieck group of constructible etale motives, the image of the nearby motive associated with a flat morphism of $$k$$ -schemes $$f:X\rightarrow \mathbb A ^1_k$$ , in the sense of Ayoub’s theory, can be identified with the image of Denef and Loeser’s motivic nearby cycles associated with $$f$$ . In particular, it provides a realization of the motivic Milnor fiber of $$f$$ in the “non-virtual” motivic world.

Trialitarian automorphisms of lie algebras

Over an algebraically closed field of characteristic zero simple Lie algebras admit outer automorphisms of order 3 if and only if they are of type D4. Moreover, thereare two conjugacy classes of such automorphisms. Among orthogonal Lie algebras over arbitrary fields of characteristic zero, only orthogonal Lie algebras relative to quadratic norm forms of Cayley algebras admit outer automorphisms of order 3. We give a complete list of conjugacy classes of outer automorphisms of order 3 for orthogonal Lie algebras over arbitrary fields of characteristic zero. For the norm form of a given Cayley algebra, one class is associated with the Cayley algebra and the others with central simple algebras of degree 3 with involution of the second kind such that the cohomological invariant of the involution is the norm form.

Quadratic and cubic invariants of unipotent affine automorphisms

Let K be an arbitrary field of characteristic zero, P n : = K [ x 1 , … , x n ] be a polynomial algebra, and P n , x 1 : = K [ x 1 −1 , x 1 , … , x n ] , for n ⩾ 2 . Let σ ′ ∈ Aut K ( P n ) be given by x 1 ↦ x 1 − 1 , x 2 ↦ x 2 + x 1 , … , x n ↦ x n + x n − 1 . It is proved that the algebra of invariants, F n ′ : = P n σ ′ , is a polynomial algebra in n − 1 variables which is generated by [ n 2 ] quadratic and [ n − 1 2 ] cubic (free) generators that are given explicitly. Let σ ∈ Aut K ( P n ) be given by x 1 ↦ x 1 , x 2 ↦ x 2 + x 1 , … , x n ↦ x n + x n − 1 . It is well known that the algebra of invariants, F n : = P n σ , is finitely generated (theorem of Weitzenböck [R. Weitzenböck, Über die invarianten Gruppen, Acta Math. 58 (1932) 453–494]), has transcendence degree n − 1 , and that one can give an explicit transcendence basis in which the elements have degrees 1 , 2 , 3 , … , n − 1 . However, it is an old open problem to find explicit generators for F n . We find an explicit vector space basis for the quadratic invariants, and prove that the algebra of invariants P n , x 1 σ is a polynomial algebra over K [ x 1 , x 1 −1 ] in n − 2 variables which is generated by [ n − 1 2 ] quadratic and [ n − 2 2 ] cubic (free) generators that are given explicitly. The coefficients of these quadratic and cubic invariants throw light on the ‘unpredictable combinatorics’ of invariants of affine automorphisms and of SL 2 -invariants.

Combinatorial geometries of field extensions

We classify the algebraic combinatorial geometries of arbitrary field extensions of transcendence degree greater than 4 and describe their groups of automorphisms. Our results and proofs extend similar results and proofs by Evans and Hrushovski in the case of algebraically closed fields. The classification of projective planes in algebraic combinatorial geometries in arbitrary fields of characteristic zero will also be given.

Affine surfaces with trivial Makar-Limanov invariant

We study the class of 2-dimensional affine k-domains R satisfying ML ( R ) = k , where k is an arbitrary field of characteristic zero. In particular, we obtain the following result: Let R be a localization of a polynomial ring in finitely many variables over a field of characteristic zero. If ML ( R ) = K for some field K ⊂ R such that trdeg K R = 2 , then R is K-isomorphic to K [ X , Y , Z ] / ( X Y − P ( Z ) ) for some nonconstant P ( Z ) ∈ K [ Z ] .

Reduction theorems for Clifford classes

Clifford Theory concerns the representations over a field F of a finite group G and one of the normal subgroups N of G. It is often viewed as a series of reduction theorems. Clifford classes are certain equivalence classes of G/N-algebras over F. They have been used to describe the Schur indices of the irreducible characters of certain families of classical groups. In the present paper, we show how Clifford classes can also be used to explain and to reflect the reduction theorems of classical Clifford theory. We show that certain products of characters correspond to certain products of Clifford classes. We show that induction, restriction, inflation, and extension of the base field can all be defined naturally for Clifford classes. We prove that the properties of these elementary operations on Clifford classes imply all of the standard Clifford-theoretic reductions in the case when the field is ℂ. They also provide important information in the case when the field F is an arbitrary field of characteristic zero.

Existence of triangular Lie bialgebra structures II

We characterize finite-dimensional Lie algebras over an arbitrary field of characteristic zero which admit a non-trivial (quasi-) triangular Lie bialgebra structure.

Lichtenbaum-Tate duality for varieties over p-adic fields

S. Lichtenbaum has proved in [L1] that there is a nondegenerate pairing Pic(C)x Br(C)->Br(K) =Q/Z (1) between the Picard group and the Brauer group of a nonsingular projective curve C over a p-adic field K (a finite extension of the p-adic numbers Qp). His proof consists of a reduction via explicit cocycle calculations in Galois cohomology to a combination of Tate duality for group schemes over p-adic fields and the autoduality of the Jacobian of a smooth curve. In this paper we will reconstruct the above duality as a purely formal combination of a generalized form of Tate duality over p-adic fields and a form of Poincar´ e duality for curves over arbitrary fields of characteristic zero. This gives a more conceptual proof of Lichtenbaum's result and an analogue in higher dimensions.

Normal Parametrizations of Algebraic Plane Curves

In this paper we present a theoretical and algorithmic analysis on the normality of rational parametrizations of algebraic plane curves over arbitrary fields of characteristic zero. If the field is algebraically closed we give an algorithm to decide whether a parametrization is proper and, if not, a normal parametrization is computed. If the field is not algebraically closed the problem is more complicated, and a degenerated situation may appear. We classify the degenerations in strong and weak degenerations, and an algorithm to decide this phenomenon is derived. Furthermore, we prove that if the parametrization is strongly degenerated then the curve cannot be normally parametrized, but weak degenerations can be resolved, and an algorithm to reparametrize the input weakly degenerated parametrization into a non-degenerated one is given. In addition, we show how these results can be applied and improved to the case of real rational curves. In this case, we present an algorithm that decides whether a given real parametrization can be normally parametrized over the reals, and that computes such a parametrization if it exists.