New effective bounds on the dimension of a linear system in $\mathbb P^2$

典范 $\mathbb{Q}$-Fano 三维流形的反典范体积的有效上界

An effective upper bound for anti-canonical volumes of canonical $\mathbb{Q}$-Fano threefolds

本文证明了典范弱 $\mathbb{Q}$-Fano 三维流形的反典范体积上界为 $324$，给出了有效估计。

Boundedness of $\mathbb{Q}$-Fano varieties with degrees and alpha-invariants bounded from below

We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.

Some uniform bounds for elliptic curves over $\mathbb Q$

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves $E/\mathbb Q$. We consider in particular the subgroup of scalars in the image of Galois, the first Galois cohomology group with values in the torsion of $E$, and the Kummer extensions generated by points of infinite order in $E(\mathbb Q)$.

The global dimension of the algebras of polynomial integro-differential operators $\mathbb{I}_n$ and the Jacobian algebras $\mathbb{A}_n$

The aim of the paper is to prove two conjectures that the (left and right) global dimension of the algebra of polynomial integro-differential operators $\mathbb{I}_n$ and the Jacobian algebra $\mathbb{A}_n$ is equal to $n$ (over a field of characteristic zero). An analogue of Hilbert's Syzygy Theorem is proven for them. The algebras $\mathbb{I}_n$ and $\mathbb{A}_n$ are neither left nor right Noetherian. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. It is proven that the global dimension of all prime factor algebras of the algebras $\mathbb{I}…

从 $\mathbb B^2$ 到 $\mathbb B^4$ 的有理真映射的锐利次数界

Sharp degree bound for rational proper maps from $\mathbb B^2$ to $\mathbb B^4$

本文证明了从二维复球到四维复球的有理真映射的D'Angelo次数猜想，通过特征数方法确立五次为锐利上界。

On a conjecture of Demailly and new bounds on Waldschmidt constants in ${\mathbb P}^N$

In the present note we prove a conjecture of Demailly for finite sets of sufficiently many very general points in projective spaces. This gives a lower bound on Waldschmidt constants of such sets. Waldschmidt constants are asymptotic invariants of subschemes receiving recently considerable amount of attention.