Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the existence of a Gysin morphism for the Blow-up of an ordinary singularity
103
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $theta in Hom_{D(Y)}(Rpi_*mathbb Q_X, mathbb Q_Y)$, compatible with pullback and with restriction to the complement of the singularity.
rate research
Read More
We propose a general definition of mathematical instanton bundle with given charge on any Fano threefold extending the classical definitions on $mathbb P^3$ and on Fano threefold with cyclic Picard group. Then we deal with the case of the blow up of $mathbb P^3$ at a point, giving an explicit construction of instanton bundles satisfying some important extra properties: moreover, we also show that they correspond to smooth points of a component of the moduli space.
We prove a monomialization theorem for mappings in general classes of infinitely differentiable functions that are called quasianalytic. Examples include Denjoy-Carleman classes (of interest in real analysis), the class of infinitely differentiable functions which are definable in a given polynomially bounded o-minimal structure (in model theory), as well as the classes of real- or complex-analytic functions, and algebraic functions over any field of characteristic zero. The monomialization theorem asserts that a mapping in a quasianalytic class can be transformed to a mapping whose components are monomials with respect to suitable local coordinates, by sequences of simple modifications of the source and target (local blowings-up and power substitutions in the real cases, in general, and local blowings-up alone in the algebraic or analytic cases). Monomialization is a version of resolution of singularities for a mapping. It is not possible, in general, to monomialize by global blowings-up, even in the real analytic case.
We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $phi colon X to mathbb{P}^n$ such that $X$ is geometrically irreducible and the nonempty fibers of $phi$ all have the same dimension, the locus of hyperplanes $H$ such that $phi^{-1} H$ is not geometrically irreducible has dimension at most $operatorname{codim} phi(X)+1$. We give an application to monodromy groups above hyperplane sections.
We study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density $rho(x)$ and a power-like reaction term. We show that for small enough initial data, if $rho(x)sim frac{1}{left(log|x|right)^{alpha}|x|^{2}}$ as $|x|to infty$, then solutions globally exist for any $p>1$. On the other hand, when $rho(x)simfrac{left(log|x|right)^{alpha}}{|x|^{2}}$ as $|x|to infty$, if the initial datum is small enough then one has global existence of the solution for any $p>m$, while if the initial datum is large enough then the blow-up of the solutions occurs for any $p>m$. Such results generalize those established in [27] and [28], where it is supposed that $rho(x)sim |x|^{-q}$ for $q>0$ as $|x|to infty$.
We consider the nonlinear heat equation with a nonlinear gradient term: $partial_t u =Delta u+mu| abla u|^q+|u|^{p-1}u,; mu>0,; q=2p/(p+1),; p>3,; tin (0,T),; xin R^N.$ We construct a solution which blows up in finite time $T>0.$ We also give a sharp description of its blow-up profile and show that it is stable with respect to perturbations in initial data. The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. The blow-up profile does not scale as $(T-t)^{1/2}|log(T-t)|^{1/2},$ like in the standard nonlinear heat equation, i.e. $mu=0,$ but as $(T-t)^{1/2}|log(T-t)|^{beta}$ with $beta=(p+1)/[2(p-1)]>1/2.$ We also show that $u$ and $ abla u$ blow up simultaneously and at a single point, and give the final profile. In particular, the final profile is more singular than the case of the standard nonlinear heat equation.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
