Related papers: The log canonical threshold and rational singulari…
We study the Hilbert series of a family of ideals J_\phi generated by powers of linear forms in k[x_1,...,x_n]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints in P^{n-1}. In the three variable case…
Recent work ([18], [1]) has produced a complete list of weighted homogeneous surface singularities admitting smoothings whose Milnor fibre has only trivial rational homology (a "rational homology disk"). Though these special singularities…
Given a map $\phi:X\rightarrow Y$ between $F$-analytic manifolds over a local field $F$ of characteristic $0$, we introduce an invariant $\epsilon_{\star}(\phi)$ which quantifies the integrability of pushforwards of smooth compactly…
Consider a rational map $f$ of degree at least 2 acting on its Julia set $J(f)$, a H\"older continuous potential $\phi: J(f)\rightarrow \R$ and the pressure $P(f,\phi). In the case where $\sup_{J(f)}\phi<P(f,phi)$, the uniqueness and…
We associate to every graph a linear program for packings of vertex disjoint paths. We show that the optimal primal and dual values of the corresponding integer program are the binomial grade and height of the binomial edge ideal of the…
A resolution-free definition of rational singularities is introduced, and it is proved that for a variety admitting a resolution of singularities, so in particular in characteristic zero, this is equivalent to the usual definition. It is…
Let $H$ be a division ring of finite dimension over its center, let $H[T]$ be the ring of polynomials in a central variable over $H$, and let $H(T)$ be its quotient skew field. We show that every intermediate division ring between $H$ and…
In this paper, we prove that if a $3$-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>3$ has only log canonical singularities, then so does a general hyperplane section $H$ of $X$. We also…
We review some basic facts on vector fields, in the complex-analytic setting, thus, obtaining a rationality result and an extension of the Birkhoff-Grothendieck theorem, as follows: (1) Let $Z$ be a compact complex manifold endowed with a…
We prove a few uniform versions of the Mordell-Lang Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points. The main focus is on function fields having high transcendence degree over the…
The extended L\"uroth's Theorem says that if the transcendence degree of $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK$ is 1 then there exists $f \in \KK(\underline{X})$ such that $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)$ is equal to $\KK(f)$. In…
If $f$ is a transcendental entire function with only algebraic singularities we calculate the Ruelle operator of $f$. Moreover, we prove both (i) if $f$ has a summable critical point, then $f$ is not structurally stable under certain…
The log canonical threshold (lct) is a fundamental invariant in birational geometry, essential for understanding the complexity of singularities in algebraic varieties. Its real counterpart, the real log canonical threshold (rlct), also…
We study the \L ojasiewicz exponent and the log canonical threshold of ideals of $\mathcal O_n$ when restricted to generic subspaces of $\mathbb C^n$ of different dimensions. We obtain effective formulas of the resulting numbers for ideals…
Brjuno and R\"ussmann proved that every irrationally indifferent fixed point of an analytic function with a Brjuno rotation number is linearizable, and Yoccoz proved that this is sharp for quadratic polynomials. Douady conjectured that this…
Let $f$ be an isolated singularity at the origin of $\mathbb{C}^n$. One of many invariants that can be associated with $f$ is its {\L}ojasiewicz exponent $\mathcal{L}_0 (f)$, which measures, to some extent, the topology of $f$. We give, for…
There are two fundamental problems motivated by Silverman's conversations over the years concerning the nature of the exact values of canonical heights of $f(z)\in\bar{\mathbb{Q}}(z)$ where $f$ has degree $d\geq 2$. The first problem is the…
A smooth hypersurface over a finite field $\mathbb{F}_q$ is called Frobenius nonclassical if the image of every geometric point under the $q$-th Frobenius endomorphism remains in the unique hyperplane tangent to the point. In this paper, we…
Let $I$ be an equidimensional ideal of a ring polynomial $R$ over $\mathbb{C}$ and let $J$ be its generic linkage. We prove that there is a uniform bound of the difference between the F-pure thresholds of $I_p$ and $J_p$ via the generalized…
We characterize the ideals $I$ of $\mathcal O_n$ of finite colength whose integral closure is equal to the integral closure of an ideal generated by pure monomials. This characterization, which is motivated by an inequality proven by…