Ivan Cheltsov, Jihun Park
We classify all pencils on a general weighted hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ whose general members are surfaces of Kodaira dimension zero.
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Stability of pencils of plane sextics and Halphen pencils of index two
本研究从几何不变量理论角度探讨平面六次曲线束的稳定性，给出了指数为2的Halphen束稳定性的完整几何刻画，丰富了代数曲线束稳定性的研究成果。
For every smooth quartic threefold, we classify all pencils on it whose general element is an irreducible surface birational to a smooth surface of Kodaira dimension zero.
Explicit Constructions of Halphen Pencils
该研究通过在射影平面中显式构造Halphen线束，得到指数为2的有理椭圆曲面，给出各类奇异纤维的实例（部分类型给出全部实例），建立了曲面纤维与平面曲线的对应并研究了线束中曲线的奇点。
We prove a criterion of nonsingularity of a complete intersection of two fiberwise quadrics in a scroll over $P^1$. As a corollary we derive the following addition to the Alexeev theorem on rationality of standard Del Pezzo fibrations of degree 4 over $P^1$: we prove that any fibration of this kind with the topological Euler characteristic -4 is rational.
We give a description of the relative Hilbert scheme of lines in the Dwork pencil of quintic threefolds. We describe the corresponding relative Hilbert scheme associated to the mirror family of quintic threefolds.
In this paper, the Euler characteristic formula for projective logarithmic minimal degenerations of surfaces with Kodaira dimension zero over a 1-dimensional complex disk is proved under a reasonable assumption and as its application, the singularity of logarithmic minimal degenerations are determined in the abelian or hyperelliptic case. By globalizing this local analysis of singular fibres via generalized canonical bundle formulae due to Fujino-Mori, we bound the number of singular fibres of abelian fibred Calabi-Yau threefolds from above,which was previously done by Oguiso in the potentiall…
The quartic hypersurfaces in P^4 invariant under the standard representation of S_6 form a linear pencil. We prove that a general member of this pencil is not rational.
Geometry of Prym semicanonical pencils and an application to cubic threefolds
本研究探讨模空间中半典范线束对应除子上的Prym映射，揭示其差异与低亏格几何特性，并通过对特定除子的分析得到三次三维簇直线的计数结果。
I construct some smooth Calabi-Yau threefolds in characteristic two and three that do not lift to characteristic zero. These threefolds are pencils of supersingular K3-surfaces. The construction depends on Moret-Bailly's pencil of abelian surfaces and Katsura's analysis of generalized Kummer surfaces. The threefold in characteristic two turns out to be nonrigid.
We describe pretty examples of derived equivalences and autoequivalences of Calabi-Yau threefolds arising from pencils of cubic fourfolds. The cubic fourfolds are chosen to be special, so they have an associated K3 surface. Thus a pencil gives rise to two different Calabi-Yau threefolds: the associated pencil of K3 surfaces, and the baselocus of the original pencil - the intersection of two cubic fourfolds. They both have crepant resolutions which are derived equivalent.
On the GIT stability of linear systems of hypersurfaces in projective space
该研究借助几何不变量理论（GIT）解决射影空间中固定次数超曲面线性系统的射影等价分类问题，给出显式判据，并应用于平面三次曲线束、任意指数Halphen束的GIT稳定性分析。
Manin involutions for elliptic pencils and discrete integrable systems
本文研究了平面双有理映射生成的离散可积系统的代数几何，给出了高次椭圆纤维丛的Manin对合几何描述，并刻画了基点特殊几何特征以确保Manin对合复合构成低次可积映射，同时将部分可积Kahan离散化识别为Manin对合的复合。
We study certain pencils of del Pezzo surfaces generated by a smooth del Pezzo surface S of degree less or equal to 3 anti-canonically embedded into a weighted projective space P and an appropriate multiple of a hyperplane H. Our main observation is that every minimal model program relative to the morphism lifting such pencil on a suitable resolution of its indeterminacies preserves the open subset P H â A^3. As an application, we obtain projective completions of A^3 into del Pezzo fibrations over P^1 of every degree less or equal to 4. We also obtain completions of A^3 into Mori conic bundles…
We present an explicit parametrization of the families of lines of the Dwork pencil of quintic threefolds. This gives rise to isomorphic curves which parametrize the lines. These curves are 125:1 covers of certain genus six curves. These genus six curves are first presented as curves in P^1*P^1 that have three nodes. It is natural to blow up P^1*P^1 in the three points corresponding to the nodes in order to produce smooth curves. The result of blowing up P^1*P^1 in three points is the quintic del Pezzo surface dP_5, whose automorphism group is the permutation group S_5, which is also a symmetr…
The paper is joined with arXiv:0911.5428 and improved. We prove that Landau-Ginzburg models for all 17 smooth Fano threefolds with Picard rank 1 can be represented as Laurent polynomials in 3 variables exhibiting them case by case. We check that these Landau-Ginzburg models can be compactified to open Calabi-Yau varieties. In the spirit of L. Katzarkov's program we prove that numbers of irreducible components of the central fibers of compactifications of these pencils are dimensions of intermediate Jacobians of Fano varieties plus 1. In particular these numbers do not depend on compactificatio…
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