Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e=[e_1,...,e_l] be a partition of n. Denote by X^e the set of l-tuples (P_1,...,P_l) of distinct nonsingular curvilinear infinitely near points of X of orders (e_1,...,e_l). We show that the group Aut(X) acts transitively on X^e. This statement generalizes earlier work where the case of the trivial partition e=[1,...,1] was treated under the supplementary condition that X is nonsingular. As an application we classify singular real rational surfaces obtained from nonsingular surfaces by performing weighted blow-ups.
In this paper we apply Shokurov's inductive method to study terminal and canonical singularities. As an easy consequence of the Minimal Model Program we show that for any three-dimensional log terminal singularity there exists some special, so called, plt blow-up. We discuss properties of them and construct some examples.
We address the question of the classification under blow-Nash equivalence of simple Nash function germs. We state that this classification coincides with the real analytic classification. We prove moreover that a simple germ can not be blow-Nash equivalent to a nonsimple one. The method is based on the computation of relevant coefficients of the real zeta functions associated to a Nash germ via motivic integration.
A recent paper of Bresciani and Vistoli studies fields of moduli using residual gerbes. One of their applications is to twisted forms of quotient singularities. We discuss a direct approach to automorphisms and twisted forms, which also works in many wildly ramified cases.
The paper completes the study of symmetries of parabolic function singularities with relation to complex crystallographic groups that was started in \cite{GM,X9}. We classify smoothable automorphisms of $P_8$ singularities which split the kernel of the intersection form on the second homology. For such automorphisms, the monodromy groups acting on the duals to the eigenspaces with degenerate intersection form are then identified as some of complex affine reflection groups tabled in \cite{P}.
Given a compact Riemann surface $X$ of genus at least $2$ with automorphism group $G$ we provide formulae that enable us to compute traces of automorphisms of X on the space of global sections of $G$-linearized line bundles defined on certain blow-ups of proyective spaces along the curve $X$. The method is an adaptation of one used by Thaddeus to compute the dimensions of those spaces. In particular we can compute the traces of automorphisms of $X$ on the Verlinde spaces corresponding to the moduli space $SU_{X}(2,ξ)$ when $ξ$ is a line bundle $G$-linearized of suitable degree.
We discuss automorphisms and pseudo-automorphisms on blowups of complex projective space with an eye to finding ones with interesting dynamical behavior.
In this paper we give a necessary combinatorial condition for a negative--definite plumbing tree to be suitable for rational blow--down, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing. New examples of surface singularities with rational homology disk smoothings are also presented; these include singularities with resolution graph having valency four nodes.
We extend the Cox-Hu-Keel construction of the Cox rings to any proper birational morphisms of normal noetherian schemes. It allows the representation of any proper birational morphism by a map of schemes with mild singularities with torus actions. In a particular case, the notion generalizes the combinatorial construction of Satriano and the recent construction of multiple weighted blow-ups on Artin-stacks by Abramovich-Quek. The latter can be viewed as an extension of stack theoretic blow-ups by Abramovich, Temkin, and Wlodarczyk, a similar construction of McQuillan and the author's recent co…
Let $X_0$ be a smooth projective threefold which is Fano or which has Picard number $1$. Let $π:X\rightarrow X_0$ be a finite composition of blowups along smooth centers. We show that for "almost all" of such $X$, if $f\in Aut(X)$ then its first and second dynamical degrees are the same. We also construct many examples of finite blowups $X\rightarrow X_0$, on which any automorphism is of zero entropy. The main idea is that because of the log-concavity of dynamical systems and the invariance of Chern classes under holomorphic automorphisms, there are some constraints on the nef cohomology class…
If a smooth projective threefold $X$ satisfies a certain Property A (see below for definition), then any automorphism of $X$ has zero entropy. Let $Y$ be a smooth projective threefold satisfying Property A. Let $π:X\rightarrow Y$ be a blowup at either a point or at a smooth curve $C\subset Y$ with the following two properties: i) $c_1(Y).C$ is an odd number, and ii) the normal vector bundle $N_{C/Y} $ is decomposable. Then we show that $X$ also satisfies Property A. As a further application of Property A we prove the following result. Let $X_1$ be the blowup of $X_0=\mathbb{P}^3$ at a finite n…
We show that the minimum weight of a weighted blow-up of $\mathbf A^d$ with $\varepsilon$-log canonical singularities is bounded by a constant depending only on $\varepsilon $ and $d$. This was conjectured by Birkar. Using the recent classification of $4$-dimensional empty simplices by Iglesias-Valiño and Santos, we work out an explicit bound for blowups of $\mathbf A^4$ with terminal singularities: the smallest weight is always at most $32$, and at most $6$ in all but finitely many cases.
In this paper we present new proofs using real spectra of the finiteness theorem on Nash trivial simultaneous resolution and the finiteness theorem on Blow-Nash triviality for isolated real algebraic singularities. That is, we prove that a family of Nash sets in a Nash manifold indexed by a semialgebraic set always admits a Nash trivial simultaneous resolution after a partition of the parameter space into finitely many semialgebraic pieces and in the case of isolated singularities it admits a finite Blow-Nash trivialization. We also complement the finiteness results with recursive bounds.
We survey some recent progress in the study of algebraic varieties X with log terminal singularities, especially, the uni-ruledness of the smooth locus X^0 of X, the fundamental group of X^0 and the automorphisms group on (smooth or singular) X when dim X = 2. The full automorphism groups of a few interesting types of K3 surfaces are described, mainly by Keum-Kondo. We conjecture that when X is Q-Fano then X^0 has a finite fundamental group, which had been proved if either dim X < 3 or the Fano index is bigger than dim X - 2. We also conjecture that when X is a log Enriques (e.g. a normal K3 o…
We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups.
Let f: X -> Z be a local, projective, divisorial contraction between normal varieties of dimension n with Q-factorial singularities. Let $Y \subset X$ be a f-ample Cartier divisor and assume that f|Y: Y -> W has a structure of a weighted blow-up. We prove that f: X -> Z, as well, has a structure of weighted blow-up. As an application we consider a local projective contraction f: X -> Z from a variety X with terminal Q-factorial singularities, which contracts a prime divisor E to an isolated Q-factorial singularity $P\in Z$, such that $-(K_X + (n-3)L)$ is f-ample, for a f-ample Cartier divisor …
The blow-up of a singularity at the module of derivations
Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México（墨西哥国立自治大学库埃纳瓦卡数学研究所）；Instituto de Ingeniería y Tecnología, UACJ（奇瓦瓦自治科技大学工程与技术研究所）
We show that if $ϕ: X \to X$ is an automorphism of a smooth projective variety and $D \subset X$ is an irreducible divisor for which the set of $d$ in $D$ with $ϕ^n(d)$ in $D$ for some nonzero $n$ is not Zariski dense, then $(X, ϕ)$ admits an equivariant rational fibration to a curve. As a consequence, we show that certain blowups (e.g. blowups in high codimension) do not alter the finiteness of $\textrm{Aut}(X)$, extending results of Bayraktar-Cantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety …
From the degree zero part of logarithmic vector fields along an algebraic hypersurface singularity we indentify the maximal multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that all such maximal tori are conjugate and in one-to-one correspondence to maxmimal tori in the degree zero jet of the embedded automorphism group. The result is motivated by Kyoji Saito's characterization of quasihomogeneity for isolated hypersurface singularities and extends its formal version and a result of Hauser and Mueller.
We establish an algorithm for resolution of singularities of an idealistic filtration in dimension 3 (at the local level) in positive characteristic, incorporating the method recently developed by Benito-Villamayor into our framework. Although (a global version of) our algorithm only implies embedded resolution of surfaces in the smooth ambient space of dimension 3, a classical result known before, we introduce some new invariant which effectively measures how much singularities are improved in the process of our algorithm and which strictly drops after each blow up. This is in contrast to the…
In this article, we prove that any complex smooth rational surface $X$ which has no automorphism of positive entropy has a finite number of real forms (this is especially the case if $X$ cannot be obtained by blowing up $\mathbb P^2_{\mathbb C}$ at $r\geq 10$ points). In particular, we prove that the group $\mathrm{Aut}^{\#}X$ of complex automorphisms of $X$ which act trivially on the Picard group of $X$ is a linear algebraic group defined over $\mathbb R$.
Let $\overline{\mathcal{M}}_{g,A[n]}$ be the moduli stack parametrizing weighted stable curves, and let $\overline{M}_{g,A[n]}$ be its coarse moduli space. These spaces have been introduced by B. Hassett, as compactifications of $\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively, by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \leq 1$ to the markings. In particular, the classical Deligne-Mumford compactification arises for $a_1 = ... = a_n = 1$. In genus zero some of these spaces appear as intermediate steps of the blow-up construction of $\overline{M}_{0,n}$ developed by M. Kapra…
In this paper we characterize the Blowing-up maps of ordinary singularities for which there exists a natural Gysin morphism, i.e. a bivariant class $θ\in Hom_{D(Y)}(Rπ_*\mathbb Q_X, \mathbb Q_Y)$, compatible with pullback and with restriction to the complement of the singularity.
We prove that the set of singular configurations of a general Gough Stewart platform has a rational parametrization. We introduce a reciprocal twist mapping which, for a general orientation of the platform, realizes the cubic surface of singularities as the blowing up of a quadric surface in five points.
We construct complete Kahler metrics of Saper type on the nonsingular set of a subvariety X of a compact Kahler manifold using (a) a method for replacing a sequence of blow-ups along smooth centers, used to resolve the singularities of X, with a single blow-up along a product of coherent ideals corresponding to the centers and (b) an explicit local formula for a Chern form associated to this single blow-up. Our metrics have a particularly simple local formula, involving essentially a product of distances to the centers of the blow-ups used to resolve the singularities of X. Our proof of (a) us…
In this paper we study $(ε,δ)$-lc singularites, i.e. $ε$-lc singularities admitting a $δ$-plt blow-up. We prove that $n$-dimensional $(ε,δ)$-lc singularities are bounded up to a deformation, and $2$-dimensional $(ε,δ)$-lc singularities form a bounded family. Furthermore, we give an example which shows that $(ε,δ)$-lc singularities are not bounded in higher dimensions, even in the analytic sense.
We construct k-parameter families of rational surface automorphisms for any k. These are automorphisms of surfaces X, which are constructed from iterated blowups over the projective plane. In certain cases: we are able to determine the exact automorphism group of X, as well as when two of the surfaces X are inequivalent.
A rational surface with discrete and non-finitely generated automorphism group
National University of Singapore（新加坡国立大学）；the University of Tokyo（东京大学）；National Center for Theoretical Sciences, Mathematics Division, National Taiwan University（台湾大学数学理论科学中心）；Tianjin University（天津大学）
We construct some positive entropy automorphisms of rational surfaces with no periodic curves. The surfaces in question, which we term tri-Coble surfaces, are blow-ups of the projective plane at 12 points which have contractions down to three different Coble surfaces. The automorphisms arise as compositions of lifts of Bertini involutions from certain degree 1 weak del Pezzo surfaces.
We prove that there is a projective K3 surface admitting a (fixed point) free automorphism of positive entropy and that no smooth compact Kähler surface other than projective K3 surfaces and their blow up admits such an automorphism.
In this note, we prove that every automorphism of a rational manifold which is obtained from $\Bbb{P}^k$ by a finite sequence blow-ups along smooth centers of dimension at most r with k>2r+2 has zero topological entropy.
We recall the construction of triality automorphism of so(8) given by E. Cartan and we give a matrix representation for the real form so(4,4). We compute the induced results on the characteristic classes. Paralelly we study the triality automorphism of the singularity D4 (in Arnolds classification of smooth functions) and its miniversal deformation. The similarity with Lie theory leads us to a definition of G2 singularity.
Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；Stony Brook University（石溪大学）；University of California, Davis（加州大学戴维斯分校）
We compute the Nekrasov partition function of gauge theories on the (resolved) toric singularities C^2/Γin terms of blow-up formulae. We discuss the expansion of the partition function in the ε_1,ε_2 \to 0 limit along with its modular properties and how to derive them from the M-theory perspective. On the two-dimensional conformal field theory side, our results can be interpreted in terms of representations of the direct sum of Heisenberg plus W_N-algebrae with suitable central charges, which can be computed from the fan of the resolved toric variety.We provide a check of this correspondence b…
Suppose that $C=(C_1,..., C_m)$ is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold $(X,ω)$ with connected, negative definite intersection graph $Γ_C$. We show that by replacing an appropriate neighborhood of $\cup C_i$ with a smoothing $W_S$ of a normal surface singularity $(S, 0)$ with resolution graph $Γ_C$, the resulting 4-manifold admits a symplectic structure. This operation generalizes the rational blow-down operation of Fintushel-Stern for other configurations, and therefore extends Symington's result about symplectic rational blow-downs.
Formal splitting and stack-theoretic normal crossings desingularization
Université Paris Cité and Sorbonne Université, UFR de Mathématiques, Institut de Mathématiques de Jussieu-Paris Rive Gauche（巴黎西岱大学与索邦大学，数学学院，儒勒·让苏-巴黎左岸数学研究所）；Institut universitaire de France (IUF)（法国大学研究院）；University of Toronto, Department of Mathematics（多伦多大学，数学系）
We give a method for constructing many examples of automorphisms with positive entropy on rational complex surfaces. The general idea is to begin with a quadratic Cremona transformation that fixes a reduced cubic curve and then use the group structure on the cubic to understand when the indeterminacy and exceptional behavior of the transformation may be eliminated by repeated blowing up.
