We give examples of degree functions deg : R --> M, where R is a polynomial ring in 2 or 3 variables and M is either the integers or the natural numbers, whose behaviour with respect to derivations D : R --> R is pathological in the sense that deg(Dx) - deg(x) (where x in R \ {0}) is not bounded above. We also give several general results stating that such pathologies do not occur when the degree functions satisfy certain hypotheses.
We prove the existence of wild automorphisms on an affine quadric threefold. The method we use is an adaptation of the one used by Shestakov and Umirbaev to prove the existence of wild automorphisms on the affine three dimensional space.
In this note we show that the set mdeg(Aut(C^3)) mdeg(Tame(C^3)) is not empty. Moreover we show that this set has infinitely many elements. Since for the famous Nagata's example N of wild automorphism, mdeg N =(5,3,1) is an element of mdeg(Tame(C^3)) and since for other known examples of wild automorphisms the multidegree is of the form (1,d_2,d_3) (after permutation if neccesary), then we give the very first exmple of wild automorphism F of C^3 such that mdeg F does not belong to mdeg(Tame(C^3)). We also show that, if d_1,d_2 are odd numbers such that gcd (d_1,d_2) =1, then (d_1,d_2,d_3) belo…
Recently, Edo-Poloni constructed a family of tame automorphisms of a polynomial ring in three variables which degenerates to a wild automorphism. In this note, we generalize the example by a different method.
We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. We give a new characterization of the finite--tame--wild trichotomy for acyclic quivers in terms of their frieze varieties. We show that an acyclic quiver is representation finite, tame, or wild, re…
A finite étale map between irreducible, normal varieties is called tame, if it is tamely ramified with respect to all partial compactifications whose boundary is the support of a strict normal crossings divisor. We prove that if the Galois group of a Galois covering contains a normal nilpotent subgroup of index bounded by a constant N, then the covering is tame if and only if it is tamely ramified with respect to a single distinguished partial compactification only depending on N. The main tools used in the proof are Temkin's local purely inseparable uniformization and a Lefschetz type theorem…
In positive characteristic, there exist counterexamples to the statement corresponding to Batyrev's theorem concerning the McKay correspondence. In this paper, we give another computation of the counterexamples by using stringy-point count for some sequences of quotient varieties. There is a proof of the Serre-Bhagava's mass formula, which is important formula in the number theory, from the computation of stringy-point count of a quotient variety associated to a representation of symmetric group. Our computation of the stringy-point count in this paper is regarded as a variation of this proof …
Suppose G is a semi-direct product of the form Z/p^n \rtimes Z/m where p is prime and m is relatively prime to p. Suppose K is a local field of characteristic p > 0. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration whose dimension depends only on the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p^3.
We provide two examples of smooth projective surfaces of tame CM type, by showing that any parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in projective 5-space is either a single point or a projective line. For surfaces of minimal degree and wild CM type, we classify rigid Ulrich bundles as Fibonacci extensions. For the rational normal scrolls S(2,3) and S(3,3), a complete classification of rigid ACM bundles is given in terms of the action of the braid group in three strands.
We study Gauss-Manin systems of non tame Laurent polynomial functions. We focuse on Hori-Vafa models, which are the expected mirror partners of the small quantum cohomology of smooth hypersurfaces in weighted projective spaces.
Let d be any number greather than or equal to 3. We show that the intersection of the set mdeg(Aut(C^3))\ mdeg(Tame(C3)) with {(d_1,d_2,d_3) : d=d_1 =< d_2 =< d_3} has infinitely many elements, where mdeg h = (deg h_1,...,deg h_n) denotes the multidegree of a polynomial mapping h=(h_1,...,h_n):C^n ---> C^n. In other words, we show that there is infiniltely many wild multidegrees of the form (d,d_2,d_3), with fixed d >= 3 and d =< d_2 =< d_3, where a sequences (d_1,...,d_n) is a wild multidegree if there is a polynomial automorphism F of C}^n with mdeg F=(d_1,...,d_n), and there is no tame autm…
We give a simpler proof as well as a generalization of the main result of an article of Shestakov and Umirbaev. This latter article being the first of two that solve a long-standing conjecture about the non-tameness, or "wildness", of Nagata's automorphism. As corollaries we get interesting informations about the leading terms of polynomials forming an automorphism in any dimension and reprove the tameness of automorphisms in dimension two.
In this paper, we study the problem of tropicalizing tame degree three coverings of the projective line. Given any degree three covering $C\longrightarrow{\mathbb{P}^{1}}$, we give an algorithm that produces the Berkovich skeleton of $C$. In particular, this gives an algorithm for finding the Berkovich skeleton of a genus $3$ curve. The algorithm uses a continuity statement for inertia groups of semistable Galois coverings, which we prove first. After that we give a formula for the decomposition group of an irreducible component $Γ\subset{\mathcal{C}_{s}}$ for a semistable Galois covering $\ma…
The purpose of the present paper is to give an effective version of the noncritical $p$-tame Belyi theorem. That is to say, we compute explicitly an upper bound of the minimal degree of tamely ramified Belyi maps in positive characteristic which are unramified at a prescribed finite set of points.
In this paper we prove an explicit formula which compares the dimensions of the spaces of vanishing cycles in a Galois cover of degree p between formal germ of curves over a complete discrete valuation ring of inequal characteristics (0,p). This formula can be easily generalised to the case of a Galois cover with group which is nilpotent or which has a normal p-Sylow subgroup. The results of this paper play a key role in [Sa-1] where is studied the semi-stable reduction of Galois covers of degree p above semi-stable curves over a complete discrete valuation ring of inequal characteristics (0,p…
Let $\mathbb X\subset\mathbb P(V)$ be a projective variety, which is not contained in a hyperplane. Then every vector $v$ in $V$ can be written as a sum of vectors from the affine cone $X$ over $\mathbb X$. The minimal number of summands in such a sum is called the rank of $v$. The set of vectors of rank $r$ is denoted by $X_r$ and its projective image by $\mathbb X_r$. The r-th secant variety of $X$ is defined $σ_r(\mathbb X):=\bar{\sqcup_{s\le r}\mathbb X_s}$; it is called tame if $σ_r(\mathbb X)=\sqcup_{s\le r} \mathbb X_s$ and wild if the closure contains elements of higher rank. In this p…
