Tropicalized quartics and canonical embeddings for tropical curves of genus 3

Bitangents of non-smooth tropical quartics

We study bitangents of non-smooth tropical plane quartics. Our main result is that with appropriate multiplicities, every such curve has 7 equivalence classes of bitangent lines. Moreover, the multiplicity of bitangent lines varies continuously in families of tropical plane curves.

组合学与热带四次曲线双切线的实提升

Combinatorics and real lifts of bitangents to tropical quartic curves

本文研究热带平面四次曲线双切线的七个等价类，给出其41种类型的完整组合分类，并建立显式符号规则以构造实提升，证实每类恰有零个或四个全实提升。

实热带四次曲线及其双切线：用补片构造计数

Real Tropical Quartics and their Bitangents: Counting with Patchworking

本文利用补片构造技术解决了特定热带四次曲线双切线实数提升行为的预测问题，并分析了实热带四次曲线的组合类型，附有polymake扩展及polyDB数据。

Buildings、赋值拟阵与热带线性空间

Buildings, valuated matroids, and tropical linear spaces

本文研究了 Affine Bruhat--Tits buildings 与赋值拟阵的联系，证明了半范数空间是所有热带化线性嵌入的极限，并验证了其作为通用可实现赋值拟阵的热带线性空间本质。

通过热带几何研究平面四次曲线的双切线：有理性、$\mathbb{A}^1$-枚举与实符号计数

Bitangents to plane quartics via tropical geometry: rationality, $\mathbb{A}^1$-enumeration, and real signed count

本文利用热带几何研究平面四次曲线双切线的算术枚举问题，给出有理性障碍和GW重数的热带刻画，并证明多数热带双切线类对$\mathbb{A}^1$-枚举贡献$2\mathbb{H}$。

K3 Polytopes and their Quartic Surfaces

K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we classify K3 polytopes with up to $30$ vertices. Their number is $36\,297\,333$. We study the singular loci of quartic surfaces that tropicalize to K3 polytopes. These surfaces are stable in the sense of Geometric Invariant Theory.

热带Weierstrass点与Weierstrass权重

Tropical Weierstrass points and Weierstrass weights

本文研究热带Weierstrass点，通过分析热带曲线上的理函数斜率，为连通分支赋予内在权重，证明了权重总和与曲线的亏格、度数和秩的关系，并探讨了热带线性序列的类比结果。

热带库默四次曲面

Tropical Kummer quartic surfaces

本文研究了热带库默四次曲面的性质及其与非阿基米德赋值域上库默四次曲面的关联，引入了有理多面体轨道群的概念并展示了其忠实嵌入。

Lifting tropical bitangents

We study lifts of tropical bitangents to the tropicalization of a given complex algebraic curve together with their lifting multiplicities. Using this characterization, we show that generically all the seven bitangents of a smooth tropical plane quartic lift in sets of four to algebraic bitangents. We do this constructively, i.e. we give solutions for the initial terms of the coefficients of the bitangent lines. This is a step towards a tropical proof that a general smooth quartic admits 28 bitangent lines. The methods are also appropriate to count real bitangents, however the conditions to de…

平面四次曲线实切线的避开轨迹与热带化

Avoidance loci and tropicalizations of real bitangents to plane quartics

本文比较了光滑平面四次曲线实切线的两种4条一组的划分方式，并利用Tarski原理、实平面四次曲线拓扑及热带几何证明二者在均有定义时完全一致。

A tropical approach to secant dimensions

Tropical geometry yields good lower bounds, in terms of certain combinatorial-polyhedral optimisation problems, on the dimensions of secant varieties. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are non-defective; this proof might be generalisable to cover all Veronese embeddings, whose secant dimensions are known from the ground-breaking but difficult work of Alexander and Hirschowitz. Also, the non-defectiveness of certain Segre embeddings is proved, w…