Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDynamical moduli spaces and polynomial endomorphisms of configurations
334
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.
rate research
Read More
Let $X$ be a quasi-projective variety and $fcolon Xto X$ a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical degree $d_1(f)$ of $f$ equals the arithmetic degree $alpha_f(P)$ at a point $P$ having Zariski dense $f$-forward orbit. Assuming $X$ is a smooth affine surface, such that the log Kodaira dimension $bar{kappa}(X)$ is non-negative (resp. the etale fundamental group $pi_1^{text{et}}(X)$ is infinite), we confirm AZO, (hence) ZDO, and KSC (when $operatorname{deg}(f)geq 2$) (resp. AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every surjective endomorphism on any projective variety with larger first dynamical degree (resp. every dominant endomorphism of any semiabelian variety).
Let $M_{g, n}$ (respectively, $overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g, n}$ (or equivalently, $overline{M_{g, n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g, n}$ is not unirational for any $n geqslant 0$ if $g geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ with $g geqslant 1$. Finding the precise range of pairs $(g, n)$, where $M_{g, n}$ is rational, stably rational or unirational, is a problem of ongoing interest. In this paper we address the rationality problem for twisted forms of $overline{M_{g, n}}$ defined over an arbitrary field $F$ of characteristic $ eq 2$. We show that all $F$-forms of $overline{M_{g, n}}$ are stably rational for $g = 1$ and $3 leqslant n leqslant 4$, $g = 2$ and $2 leqslant n leqslant 3$, $g = 3$ and $1 leqslant n leqslant 14$, $g = 4$ and $1 leqslant n leqslant 9$, $g = 5$ and $1 leqslant n leqslant 12$.
We give a summary of joint work with Michael Thaddeus that realizes toroidal compactifcations of split reductive groups as moduli spaces of framed bundles on chains of rational curves. We include an extension of this work that covers Artin stacks with good moduli spaces. We discuss, for complex groups, the symplectic counterpart of these compactifications, and conclude with some open problems about the moduli problem concerned.
An increasingly important area of interest for mathematicians is the study of Abelian differentials. This growing interest can be attributed to the interdisciplinary role this subject plays in modern mathematics, as various problems of algebraic geometry, dynamical systems, geometry and topology lead to the study of such objects. It comes as a natural consequence that we can employ in our study algebraic, analytic, combinatorial and dynamical perspectives. These lecture notes aim to provide an expository introduction to this subject that will emphasize the aforementioned links between different areas of mathematics. We will associate to an Abelian differential a flat surface with conical singularities such that the underlying Riemann surface is obtained from a polygon by identifying edges with one another via translation. We will focus on studying these objects in families and describe some properties of the orbit as we vary the polygon by the action of $GL_2^{+}(mathbb{R})$ on the plane.
We study Bridgeland moduli spaces of semistable objects of $(-1)$-classes and $(-4)$-classes in the Kuznetsov components on index one prime Fano threefold $X_{4d+2}$ of degree $4d+2$ and index two prime Fano threefold $Y_d$ of degree $d$ for $d=3,4,5$. For every Serre-invariant stability condition on the Kuznetsov components, we show that the moduli spaces of stable objects of $(-1)$-classes on $X_{4d+2}$ and $Y_d$ are isomorphic. We show that moduli spaces of stable objects of $(-1)$-classes on $X_{14}$ are realized by Fano surface $mathcal{C}(X)$ of conics, moduli spaces of semistable sheaves $M_X(2,1,6)$ and $M_X(2,-1,6)$ and the correspondent moduli spaces on cubic threefold $Y_3$ are realized by moduli spaces of stable vector bundles $M^b_Y(2,1,2)$ and $M^b_Y(2,-1,2)$. We show that moduli spaces of semistable objects of $(-4)$-classes on $Y_{d}$ are isomorphic to the moduli spaces of instanton sheaves $M^{inst}_Y$ when $d eq 1,2$, and show that therere open immersions of $M^{inst}_Y$ into moduli spaces of semistable objects of $(-4)$-classes when $d=1,2$. Finally, when $d=3,4,5$ we show that these moduli spaces are all isomorphic to $M^{ss}_X(2,0,4)$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
