Adrian Langer
We prove that the bicanonical system on a surface of general type with K^2=4 has no base components and describe clusters contracted by 4K_X for a numerical Godeaux surface and 3K_X for a numerical Campedelli surface.
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In this paper, we compute the dimensions of the 1st and 2nd cohomology groups of all the pluricanonical bundles for Hirzebruch surfaces, and the dimensions of the 1st and 2nd cohomology groups of the second pluricanonical bundles for a blow-up of a projective plane along finite many distinct points. Both of them give examples showing that the higher cohomology of the pluricanonical bundle is not deformation invariant.
We show that $|mK_X|$ defines a birational map and has no fixed part for some bounded positive integer $m$ for any $\frac{1}{2}$-lc surface $X$ such that $K_X$ is big and nef. For every positive integer $n\geq 3$, we construct a sequence of projective surfaces $X_{n,i}$, such that $K_{X_{n,i}}$ is ample, ${\rm{mld}}(X_{n,i})>\frac{1}{n}$ for every $i$, $\lim_{i\rightarrow+\infty}{\rm{mld}}(X_{n,i})=\frac{1}{n}$, and for any positive integer $m$, there exists $i$ such that $|mK_{X_{n,i}}|$ has non-zero fixed part. These results answer the surface case of a question of Xu.
In this article, we investigate some properties of cyclic coverings of complex surfaces of general type branched along smooth curves that are numerically equivalent to a multiple of the canonical class. The main results concern coverings of surfaces of general type with p_g=0 and Miyaoka--Yau surfaces; in particular, they provide new examples of multicomponent moduli spaces of surfaces with given Chern numbers as well as new examples of surfaces that are not deformation equivalent to their complex conjugates.
Stable surfaces and their log analogues are the type of varieties naturally occuring as boundary points in moduli spaces. We extend classical results of Kodaira and Bombieri to this more general setting: if $(X,Δ)$ is a stable log surface with reduced boundary (possibly empty) and $I$ is its global index, then $4I(K_X+Δ)$ is base-point-free and $8I(K_X+Δ)$ is very ample. These bounds can be improved under further assumptions on the singularities or invariants, for example, $5(K_X+Δ)$ is very ample if $(X,Δ)$ has semi-canonical singularities.
In this article we prove the following boundedness result: Fix a DCC set $I\subset [0, 1]$. Let $\mathfrak{D}$ be the set of all log pairs $(X, Δ)$ satisfying the following properties: (i) $X$ is a projective surface defined over an algebraically closed field, (ii) $(X, Δ)$ is log canonical and the coefficients of $Δ$ are in $I$, and (iii) $K_X+Δ$ is big. Then there is a positive integer $N=N(I)$ depending only on the set $I$ such that the linear system $|\lceil m(K_X+Δ)\rceil|$ defines a birational map onto its image for all $m\geq N$ and $(X, Δ)\in\mathfrak{D}$
We give a method for the computation of the plurigenera of a product-quotient manifold. We give two different types of applications to it: to the construction of Calabi-Yau threefolds and to the determination of the minimal model of a product-quotient surface of general type.
We study the flatness of log-pluricanonical sheaves on stable families of surfaces.
On generic finiteness of pluricanonical maps of threefolds of general type
本文证明了对满足P2(X)≥2的一般型极小三维流形X，|6K_X|定义通用有限映射，且该结果最优；同时证明对所有一般型极小三维流形X，当n≥38时|nK_X|定义通用有限映射。关键技术是曲面的新通用有限性准则和特殊解消下的有效比较不等式。
In this paper we study pluricanonical maps of minimal projective 3-folds of general type with geometric genus $1$, $2$ and $3$. We go in the direction pioneered by Enriques and Bombieri, and other authors, pinning down, for low projective genus, a finite list of exceptions to the birationality of some pluricanonical map. In particular, apart from a finite list of weighted baskets, we prove the birationality of $φ_{16}$, $φ_{6}$ and $φ_{5}$ respectively.
The minimal volume of stable surfaces of rank one
本文确定了秩为一的稳定表面的最小体积，并证明达到此最小值的表面在同构意义下是唯一的。该结果解决了Alexeev和第二作者的猜想。
We construct from a general del Pezzo surface of degree 1 a Gorenstein stable surfaces $X$ with $K_X^2=1$ and $p_g(X)=q(X)=0$. These surfaces are not smoothable but give an open subset of an irreducible component of the moduli space of stable Godeaux surfaces. In a particular example we also compute the canonical ring explicitly and discuss the behaviour of pluricanonical maps.
We prove a general embedding theorem for Cohen--Macaulay curves (possibly nonreduced), and deduce a cheap proof of the standard results on pluricanonical embeddings of surfaces, assuming vanishing H^1(2K_X)=0.
Birational Automorphism Bounds for General-Type Foliations on Surfaces via Pluricanonical Indices
本文针对光滑射影曲面上一般型典范叶状结构的双有理自同构群，利用多重典范指标与伴随体积的簇公式，推导得到不同 Kodaira 维数下的自同构群阶的上界。
The main result is that a quasi-projective surface has negative log Kodaira dimension (i.e. no log pluricanonical sections) iff it is dominated by images of the affine line. This follows from our main intermediate result, that the smooth locus of a log Fano surface is rationally connected. We also give a classification of all but a bounded family of rank one log del Pezzo surfaces.
On bicanonical maps of threefolds of general type with large volumes
研究一般型三维流形大体积的双CANONICAL映射的维度以及大canonical体积的pluricanonical映射的纤维结构
Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type with nef canonical divisor and canonical singularities, whose image $Y$ under the canonical map is of maximal dimension, satisfies $K_X^n \ge 2 (p_g-n)$. We investigate the very interesting extremal situation $K_X^n=2(p_g-n)$, which appears in a number of geometric situations. Since these extremal varieties are natural higher dimensional analogues of Horikawa surfaces, we name them Horikawa varieties. These varieties have been previously dealt with in…
Hirschfeld classified split del Pezzo surfaces of degree at least three whose points are all contained on the lines in the surface. We continue his work and begin the classification of split degree two del Pezzo surfaces over finite fields whose points are all on the fifty-six exceptional curves of the surfaces.
We classify Enriques involutions on a K3 surface, up to conjugation in the automorphism group, in terms of lattice theory. We enumerate such involutions on singular K3 surfaces with transcendental lattice of discriminant smaller than or equal to 36. For 11 of these K3 surfaces, we apply Borcherds method to compute the automorphism group of the Enriques surfaces covered by them. In particular, we investigate the structure of the two most algebraic Enriques surfaces.
Since the work of Ellingsrud and Peskine at the end of 1980s, it has been known that, with the exception of a finite number of families, smooth compact complex surfaces in $\mathbb{P}^4$ with prescribed Chern classes must lie on hypersurfaces of degree $m\leq 5$. The study of surfaces lying on a small degree hypersurface in $\mathbb{P}^4$---small meaning $\leq5$---seems to be a way of obtaining empirical data leading to a better conceptual understanding of surfaces in $\mathbb{P}^4$. From this perspective, two main issues are considered in the paper: - an analogue of the Hartshorne-Lichtenbaum…
For a log del Pezzo surface $S$, the fractional index $r(S)\in\mathbb{Q}_{>0}$ is the maximum of $r$ with which $-K_S$ can be written as $r$ times some Cartier divisor. We classify all the log del Pezzo surfaces $S$ with $r(S)\geq 1/2$, after the technique of Nakayama.
We establish Noether's inequality for surfaces of general type in positive characteristic.Then we extend Enriques' and Horikawa's classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.
We give a new proof of the theorem of Beauville and Voisin about the decomposition of the small diagonal of a K3 surface S. Our proof is explicit and works with the embedding of S in a projective space. It is different from the one used by Beauville and Voisin, which employed the existence of one-parameters families of elliptic curves.
We use vector-bundle techniques in order to compute $\dim W^1_d(C)$ where $C$ is general and smooth in a linear system on an unnodal Enriques surface. We furthermore find new examples of smooth curves on Enriques surfaces with an infinite number of $g^1_{\gon(C)}$'s.
We construct a surface with log terminal singularities and ample canonical class that has $K_X^2=1/48 983$ and a log canonical pair $(X,B)$ with a nonempty reduced divisor $B$ and ample $K_X+B$ that has $(K_X+B)^2 = 1/462$. Both examples significantly improve known records.
Normal surface singularities of small degrees
本文通过Yau序列推广Yau椭圆序列，利用Yau环计算二次曲面奇点的典范环，并给出算术亏格公式、几何亏格上界及加权对偶图分类性质。
Let S be a smooth, projective surface of Picard rank 1 and very ample generator embedding S into P^n. Let C be a smooth curve in O(m) for m \geq 5. We prove that any base-point free, complete g^r_d on C for r\in\{1,2\} and d small enough is cut out by a hyperplane section restricted to a multisecant (n-r-1)-plane.
Let $\mathfrak B_g$ denote the moduli space of primitively polarized $K3$ surfaces $(S,H)$ of genus $g$ over $\mathbb C$. It is well-known that $\mathfrak B_g$ is irreducible and that there are only finitely many rational curves in $|H|$ for any primitively polarized $K3$ surface $(S,H)$. So we can ask the question of finding the monodromy group of such curves. The case of $g=2$ essentially follows from the results of Harris \cite{Ha} to be the full symmetric group $S_{324}$, here we solve the case $g=3$ and $4$.
We give a recursive formula for purely real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown up at $q$ real and $s \leq 1$ pairs of conjugate imaginary points, where $q+2s\le 5$, and the real quadric blown up at $s \leq 1$ pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil's recursive formula for Gromov-Witten invariants of these surfaces and generalizes our recursive formula for purely real Welschinger invariants of real toric Del Pezzo surfaces. As a consequence, we prove the positivity of the Welschi…
Let $X$ be a surface of degree $n$, projected onto $\mathbb{CP}^2$. The surface has a natural Galois cover with Galois group $S_n.$ It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of $X.$ In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree $n \leq 4$, that degenerate to a nice plane arrangement, namely a union of $n$ planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree $4$ em…
Small instantons of exceptional groups arise geometrically by a collapsing del Pezzo surface in a CY. We use this to explain the physics of a 4-brane probe in Type I' compactification to 9 dimensions.
We give examples of local signatures, completely different from the usual ones, for general fibrations of genus $2$ and genus $3$.
We construct examples of non-isotrivial algebraic families of smooth complex projective curves over a curve of genus 2. This solves a problem from Kirby's list of problems in low-dimensional topology. Namely, we show that 2 is the smallest possible base genus that can occur in a 4-manifold of non-zero signature which is an oriented fiber bundle over a Riemann surface. A refined version of the problem asks for the minimal base genus for fixed signature and fiber genus. Our constructions also provide new (asymptotic) upper bounds for these numbers.
Shapes of surfaces that contain a great and a small circle through each point
本文针对三维单位球面中过每点都含一个大圆和一个小圆的曲面，完成了拓扑类型、实奇点轨迹分类，明确了其标准形式归属及曲面内圆与自交轨迹的相交特征。
Surfaces of general type with positive second Segre number $s_2:=c_1^2-c_2>0$ are known by results of Bogomolov to be quasi-hyperbolic i.e. with finitely many rational and elliptic curves. These results were extended by McQuillan in his proof of the Green-Griffiths conjecture for entire curves on such surfaces. In this work, we study hyperbolic properties of minimal surfaces of general type with minimal $c_1^2$, known as Horikawa surfaces. In principle these surfaces should be the most difficult case for the above conjecture as illustrate the quintic surfaces in $\bP^3$. Using orbifold techniq…
An Automorphism group of a rational surface: Not too big not too small
针对 Coxeter 群子群的自同构实现问题，构造 $W_{15}$ 中同构于 $D_3 \times \mathbb{Z}$ 的可实现子群，并指出 $W_{14}$ 中存在不可实现元素。
Extending work of M. Zarzar, we evaluate the potential of Goppa-type evaluation codes constructed from linear systems on projective algebraic surfaces with small Picard number. Putting this condition on the Picard number provides some control over the numbers of irreducible components of curves on the surface and hence over the minimum distance of the codes. We find that such surfaces do not automatically produce good codes; the sectional genus of the surface also has a major influence. Using that additional invariant, we derive bounds on the minimum distance under the assumption that the hype…
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