We provide a method to compute the dimension of the tangent space to the global infinitesimal deformation functor of a curve together with a subgroup of the group of automorphisms. The computational techniques we developed are applied to several examples including Fermat curves, $p$-cyclic covers of the affine line and to Lehr-Matignon curves.
We apply the known results on the Galois module structure of the sheaf of polydifferentials in order to study the dimension of the tangent space of the deformation functor of curves with automorphisms. We are able to find the dimension for the case of weakly ramified covers and for the case of the action of a cyclic group of order $p^v$.
We provide a coherent overview of a number of recent results obtained by the authors in the theory of schemes defined over the field with one element. Essentially, this theory encompasses the study of a functor which maps certain geometries including graphs to Deitmar schemes with additional structure, as such introducing a new zeta function for graphs. The functor is then used to determine automorphism groups of the Deitmar schemes and base extensions to fields.
We consider matrices with entries in a local ring, Mat(m,n;R). Fix an action of group G on Mat(m,n;R), and a subset of allowed deformations, Σin Mat(m,n;R). The standard question (along the lines of Singularity Theory) is the finite-(Σ,G)-determinacy of matrices. In our previous work this determinacy question was reduced to the study of the tangent spaces to Σand to the orbit, T_{(Σ,A)}, T_{(GA,A)}, and their quotient: the tangent module to the miniversal deformation. In particular, the order of determinacy is controlled by the annihilator of this tangent module. Then we have studied this tang…
We study $p$-group Galois covers $X \rightarrow \mathbb{P}^1$ with only one fully ramified point. These covers are important because of the Katz-Gabber compactification of Galois actions on complete local rings. The sequence of ramification jumps is related to the Weierstrass semigroup of the global cover at the stabilized point. We determine explicitly the jumps of the ramification filtrations in terms of pole numbers. We give applications for curves with zero $p$--rank: we focus on maximal curves and curves that admit a big action. Moreover the Galois module structure of polydifferentials is…
If $X$ is a rational surface without nonzero holomorphic vector field and $f$ is an automorphism of $X$, we study in several examples the Zariski tangent space of the local deformation space of the pair $(X, f)$.
Let $X$ be a normal projective algebraic variety, $G$ its largest connected automorphism group, and $A(G)$ the Albanese variety of $G$. We determine the isogeny class of $A(G)$ in terms of the geometry of $X$. In characteristic 0, we show that the dimension of $A(G)$ is the rank of every maximal trivial direct summand of the tangent sheaf of $X$. Also, we obtain an optimal bound for the dimension of the largest anti-affine closed subgroup of $G$ (which is the smallest closed subgroup that maps onto $A(G)$).
We show that deformations of a surjective morphism onto a Fano manifold of Picard number 1 are unobstructed and rigid modulo the automorphisms of the target, if the variety of minimal rational tangents of the Fano manifold is non-linear or finite. The condition on the variety of minimal rational tangents holds for practically all known examples of Fano manifolds of Picard number 1, except the projective space. When the variety of minimal rational tangents is non-linear, the proof is based on an earlier result of N. Mok and the author on the birationality of the tangent map. When the varieties …
We provide an explicit characterization of the covariant isotropy group of any Grothendieck topos, i.e. the group of (extended) inner automorphisms of any sheaf over a small site. As a consequence, we obtain an explicit characterization of the centre of a Grothendieck topos, i.e. the automorphism group of its identity functor.
Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic
我们计算切空间的维数，和两个变形函子的前可表示壳的克鲁尔维数。第一个是有限群G的具有给定作用的正特征域上的普通曲线X的“代数”变形函子，并根据X -> G \X的分歧行为计算数据。第二个是“解析”变形函子的一个固定的嵌入PGL（2，K）在一个非阿基米德值域K的离散群N，和数据计算的Bass-Serre表示N通过一个图形的群体。最后，如果F是N的自由子群，使得N包含在PGL（2，K）中F的正规化子中，则与F相关联的Mumford曲线具有N/F的作用，并且我们证明了使后一作用变形的代数函子与使N的嵌入变形的解析函子是一致的。
For classical dynamical systems, the polynomial entropy serves as a refined invariant of the topological entropy. In the setting of categorical dynamical systems, that is, triangulated categories endowed with an endofunctor, we develop the theory of categorical polynomial entropy, refining the categorical entropy defined by Dimitrov-Haiden-Katzarkov-Kontsevich. We justify this notion by showing that for an automorphism of a smooth projective variety, the categorical polynomial entropy of the pullback functor on the derived category coincides with the polynomial growth rate of the induced actio…
