Value distribution of derivatives in polynomial dynamics

For every $m\in \mathbb {N}$ , we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$ th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of f with pole at $\infty $ . We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on $\mathbb {P}^1(\overline {k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb {C}^2$ has a given eigenvalue.

Birational Geometry of Blow-ups of Projective Spaces Along Points and Lines

Consider the blow-up $X$ of ${\mathbb{P}}^3$ at $6$ points in very general position and the $15$ lines through the $6$ points. We construct an infinite-order pseudo-automorphism $\phi _X$ on $X$. The effective cone of $X$ has infinitely many extremal rays and, hence, $X$ is not a Mori Dream Space. The threefold $X$ has a unique anticanonical section, which is a Jacobian K3 Kummer surface $S$ of Picard number 17. The restriction of $\phi _X$ on $S$ realizes one of Keum’s 192 infinite-order automorphisms. We show the blow-up of ${\mathbb{P}}^n$ ($n\geq 3$) at $(n+3)$ very general points and certain $9$ lines through them is not a Mori Dream Space. As an application, for $n\geq 7$, the blow-up of $\overline{M}_{0,n}$ at a very general point has infinitely many extremal effective divisors.

Brill-Noether generality of binary curves

We show that the space $G^r_{\underline d}(X)$ of linear series of certain multi-degree $\underline d=(d_1,d_2)$ (including the balanced ones) and rank r on a general genus-g binary curve X has dimension $\rho _{g,r,d}=g-(r+1)(g-d+r)$ if nonempty, where $d=d_1+d_2$ . This generalizes Caporaso’s result from the case $r\leq 2$ to arbitrary rank, and shows that the space of Osserman-limit linear series on a general binary curve has the expected dimension, which was known for $r\leq 2$ . In addition, we show that the space $G^r_{\underline d}(X)$ is still of expected dimension after imposing certain ramification conditions with respect to a sequence of increasing effective divisors supported on two general points $P_i\in Z_i$ , where $i=1,2$ and $Z_1,Z_2$ are the two components of X. Our result also has potential application to the lifting problem of divisors on graphs to divisors on algebraic curves.

Fibration and classification of smooth projective toric varieties of low Picard number

In this paper, we show that a smooth toric variety [Formula: see text] of Picard number [Formula: see text] always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone [Formula: see text] of numerically effective divisors and cutting a facet of the pseudo-effective cone [Formula: see text], that is [Formula: see text]. In particular, this means that [Formula: see text] admits non-trivial and non-big numerically effective divisors. Geometrically, this guarantees the existence of a fiber type contraction morphism over a smooth toric variety of dimension and Picard number lower than those of [Formula: see text], so giving rise to a classification of smooth and complete toric varieties with [Formula: see text]. Moreover, we revise and improve results of Oda–Miyake by exhibiting an extension of the above result to projective, toric, varieties of dimension [Formula: see text] and Picard number [Formula: see text], allowing us to classifying all these threefolds. We then improve results of Fujino–Sato, by presenting sharp (counter)examples of smooth, projective, toric varieties of any dimension [Formula: see text] and Picard number [Formula: see text] whose non-trivial nef divisors are big, that is [Formula: see text]. Producing those examples represents an important goal of computational techniques in definitely setting an open geometric problem. In particular, for [Formula: see text], the given example turns out to be a weak Fano toric fourfold of Picard number 4.

Singularities of Divisors of Low Degree on Simple Abelian Varieties

It is known by the results of Kollár, Ein, Lazarsfeld, Hacon, and Debarre that effective divisors representing principal and other low-degree polarizations on complex abelian varieties have mild singularities. In this note, we extend these results to all polarizations of degree $<g$ on simple $g$-dimensional abelian varieties, settling a conjecture of Debarre and Hacon.

Effective Divisors in the Projectivized Hodge Bundle

We compute the class of the closure of the locus of canonical divisors in the projectivization of the Hodge bundle ${\mathbb{P}}\overline{{\mathcal{H}}}_g$ over $\overline{{\mathcal{M}}}_g$, which have a zero at a Weierstrass point. We also show that the strata of canonical and bicanonical divisors with a double zero span extremal rays of the respective pseudoeffective cones.

FINITENESS OF LOG MINIMAL MODELS AND NEF CURVES ON -FOLDS IN CHARACTERISTIC

In this article, we prove a finiteness result on the number of log minimal models for 3-folds in $\operatorname{char}p>5$. We then use this result to prove a version of Batyrev’s conjecture on the structure of nef cone of curves on 3-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_{X}$ of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on $X$.