Related papers: Indeterminacy loci of iterate maps in moduli space

We consider the family $\mathrm{MP}_d$ of affine conjugacy classes of polynomial maps of one complex variable with degree $d \geq 2$, and study the map $\Phi_d:\mathrm{MP}_d\to \widetilde{\Lambda}_d \subset \mathbb{C}^d / \mathfrak{S}_d$…

The iteration of rational maps is well-understood in dimension 1 but less so in higher dimensions. We study some maps on spaces of matrices which present a weak complexity with respect to the ring structure. First we give some properties of…

We study extensions of the measure of maximal entropy to suitable compactifications of the parameter space and the moduli space of rational maps acting on the Riemann sphere. For parameter space, we consider a space which resolves the…

For any given natural $d\ge 1$ we provide examples of rational self-maps of complex projective plane $\pp^2$ of degree $d$ without (holomorphic) fixed points. This makes a contrast with the situation in one dimension. We also prove that the…

We prove orientation results for evaluation maps of moduli spaces of rational stable maps to del Pezzo surfaces over a field, both in characteristic $0$ and in positive characteristic. These results and the theory of degree developed in a…

Let $K$ be a number field and $f: \mathbb{P}^1 \to \mathbb{P}^1$ a rational map of degree $d \geq 2$ with at most $s$ places of bad reduction, where we include all archimedean places. We prove that there exists constants $c_1,c_2 > 0$,…

We define a natural compactification of an arrangement complement in a ball quotient. We show that when this complement has a moduli space interpretation, then this compactification is often one that appears naturally by means of geometric…

Let $X,Y$ be two irreducible subvarieties of the projective space $\mathbb{P}^n$, and $d\geq 1$ an integer number. The main result of this paper is an algorithm to construct {\bf explicitly}, in terms of $d$ and the ideals defining $X$ and…

We investigate the topology of the space of M\"obius conjugacy classes of degree $d$ rational maps on the Riemann sphere. We show that it is rationally acyclic and we compute its fundamental group. As a byproduct, we also obtain the ranks…

The parameter space of $n$ ordered points in projective $d$-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in $(\mathbb{P}^d)^n$. The resulting variety was used to study the…

It is known that every nonorientable surface $\Sigma$ has an orientable double cover $\tilde{\Sigma}$. The covering map induces an involution on the moduli space $\tilde{\M}$ of gauge equivalence classes of flat $G$-connections on…

Consider a smooth variety $X$ and a smooth divisor $D\subset X$. Kim and Sato (arXiv:0806.3819) define a natural compactification of $(X\setminus D)^n$, denoted $X_D^{[n]}$, which is a moduli space of stable configurations of $n$ points…

There is a natural conjugation action on the set of endomorphism of $\P^N$ of fixed degree $d \geq 2$. The quotient by this action forms the moduli of degree $d$ endomorphisms of $\P^N$, denoted $\mathcal{M}_d^N$. We construct invariant…

We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space $\mathscr F_q(r, d)$ of singular foliations of codimension $q$ and degree $d$ on the complex…

In a previous paper, we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing…

We construct moduli spaces of linear self-maps of projective space with marked points, up to projective equivalence. That is, we let the special linear group act simultaneously by conjugation on projective linear maps and diagonally on…

We study the moduli space of triples $(C, L_1, L_2)$ consisting of quartic curves $C$ and lines $L_1$ and $L_2$. Specifically, we construct and compactify the moduli space in two ways: via geometric invariant theory (GIT) and by using the…

We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications of…

Let $f$ be a rational map of degree $d\geq 2$. The moduli space $\mathcal{M}_f$, introduced by McMullen and Sullivan, is a complex analytic space consisting all quasiconformal conjugacy classes of $f$. For $f$ that is not flexible Latt\`es,…

The space of smooth rational curves of degree $d$ in a projective variety $X$ has compactifications by taking closures in the Hilbert scheme, the moduli space of stable sheaves or the moduli space of stable maps respectively. In this paper…