Unbounded negativity on rational surfaces in positive characteristic

Negative curves on special rational surfaces

We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an elliptic curve and nine Fermat points. As a consequence of our analysis, we also show that the Bounded Negativity Conjecture holds for the surfaces we consider. The note contains also some problems for future attention.

Bounded Negativity and Arrangements of Lines

The Bounded Negativity Conjecture predicts that for any smooth complex surface $X$ there exists a lower bound for the selfintersection of reduced divisors on $X$. This conjecture is open. It is also not known if the existence of such a lower bound is invariant in the birational equivalence class of $X$. In the present note we introduce certain constants $H(X)$ which measure in effect the variance of the lower bounds in the birational equivalence class of $X$. We focus on rational surfaces and relate the value of $H({\mathbb P}^2)$ to certain line arrangements. Our main result is Theorem 3.3 an…

Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls

The Milnor fibre of a $\mathbb{Q}$-Gorenstein smoothing of a Wahl singularity is a rational homology ball $B_{p,q}$. For a canonically polarised surface of general type $X$, it is known that there are bounds on the number $p$ for which $B_{p,q}$ admits a symplectic embedding into $X$. In this paper, we give a recipe to construct unbounded sequences of symplectically embedded $B_{p,q}$ into surfaces of general type equipped with non-canonical symplectic forms. Ultimately, these symplectic embeddings come from Mori's theory of flips, but we give an interpretation in terms of almost toric structu…

Singular Rational Curves on Elliptic K3 Surfaces

We show that on every elliptic K3 surface $X$ there are rational curves $(R_i)_{i\in \mathbb{N}}$ such that $R_i^2 \to \infty$, i.e., of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to $\mathbb{P}(Ω_X)$ is dense in the Zariski topology. As an application we give a simple proof of a theorem of Kobayashi in the elliptic case, i.e., there are no globally defined symmetric differential forms.

Classification of planar rational cuspidal curves. II. Log del Pezzo models

Let $E\subseteq \mathbb{P}^2$ be a complex curve homeomorphic to the projective line. The Negativity Conjecture asserts that the Kodaira-Iitaka dimension of $K_X+\frac{1}{2}D$, where $(X,D)\to (\mathbb{P}^{2},E)$ is a minimal log resolution, is negative. We prove structure theorems for curves satisfying this conjecture and we finish their classification up to a projective equivalence by describing the ones whose complement admits no $\mathbb{C}^{**}$-fibration. As a consequence, we show that they satisfy the Strong Rigidity Conjecture of Flenner-Zaidenberg. The proofs are based on the almost m…

Hirzebruch曲面的非正与无穷远负除子赋值

Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces

本文研究由Hirzebruch曲面除子赋值定义的有理曲面，引入非正性与无穷远负性概念，证明其局部与全局等价条件，并在非正无穷远情形下显式给出曲线锥的极值射线。

Rigid divisors on surfaces

We study effective divisors $D$ on surfaces with $H^0(\mathcal O_D)=k$ and $H^1(\mathcal O_D)=H^0(\mathcal O_D(D))=0$. We give a numerical criterion for such divisors, following a general investigation of negativity, rigidity and connectivity properties. Examples include exceptional loci of rational singularities, and spherelike divisors.

极小有理曲面的阿贝尔典范覆盖及 $\mathbb F_1$ 至次数 $8$

Abelian canonical covers of minimal rational surfaces and $\mathbb F_1$ up to degree $8$

本文分类了极小度曲面及 $\mathbb P^2$、Hirzebruch 曲面的阿贝尔典范覆盖，完成了次数 $5$ 至 $8$ 的分类，给出了新覆盖族并计算了斜率累积点。

On the number of Mordell-Weil generators for cubic surfaces

Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal cardinality which generates S(K) via successive secant and tangent constructions. Let r(S,K) be the cardinality of such a Mordell-Weil generating set. Manin posed what is known as the Mordell-Weil problem for cubic surfaces: if K is finitely generated over its prime subfield then r(S,K) is finite. In this paper, we prove a special case of this conjecture. Namely, if S contain…

K3曲面上的曲线

Curves on K3 surfaces

本文完成了特征零下K3曲面存在无穷多有理曲线猜想的剩余情形并证明正特征下几乎所有情形，通过引入再生和标记点技巧两种形变理论新技术，证明了特征零下任意射影K3曲面上任意几何亏格的无界度整曲线的存在性。

全局迹正定非交换多项式与无界迹矩问题

Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem

本文首次建立了非交换多项式全局迹正定性的 Positivstellensatz，通过求解无界迹矩问题并结合半定优化，证明迹正定 nc 多项式可由 Hermitian 平方和与交换子显式逼近。