We study the connection between Conway-Coxeter frieze patterns and the data of the minimal resolution of a complex curve singularity: using Popescu-Pampu's notion of the lotus of a singularity, we describe a bijection between the dual resolution graphs of Newton non-degenerate plane curve singularities and Conway-Coxeter friezes. We use representation theoretic reduction methods to interpret some of the entries of the frieze coming from the partial resolutions of the corresponding curve singularity. Finally, we translate the notion of mutation, coming from cluster combinatorics, to resolutions of plane complex curves.
A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the characteristic polynomial. Here the conjecture is proved, and some of the other problems are solved, too. In the cases where also an old conjecture of Orlik on the integral monodromy holds, this has implications on the automorphism group of the Milnor lattice. The combinatorics used in the proof of the conjecture consists of tuples of orders on sets $\{0,1,.…
This paper shows that the motivic BPS invariants associated to a noncommutative crepant resolution of a compound Du-Val singularity are controlled by the labelled Dynkin combinatorics appearing in the work of Iyama--Wemyss. In particular, we show that the invariants vanish for dimension vectors which are not a multiple of a restricted root obtained from the affine root system under a natural quotient map. An immediate corollary is a description of the curve classes for which the Gopakumar--Vafa invariants of geometric crepant resolutions vanish, generalising a recent result of Nabijou--Wemyss …
This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs $(C,o)$ of complex analytic curves contained in a smooth complex analytic surface $S$. The embedded topological type of such a pair $(S, C)$ is usually defined to be that of the oriented link obtained by intersecting $C$ with a sufficiently small oriented Euclidean sphere centered at the point $o$, defined once a system of local coordinates $(x,y)$ was chosen on the germ $(S,o)$. If one works more generally over an arbitrary algebraically closed fie…
Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；Department of Mathematics, Stony Brook University（石溪大学数学系）
Understanding when an abstract complex curve of given genus comes equipped with a map of fixed degree to a projective space of fixed dimension is a foundational question; and Brill--Noether theory addresses this question via linear series, which algebraically codify maps to projective targets. Classical Brill--Noether theory, which focuses on smooth curves, has been intensively explored; but much less is known for singular curves, particularly for those with non-nodal singularities. In a one-parameter family of smooth curves specializing to a singular curve $C_0$, one expects certain aspects o…
The aim of this note is to exploit a new relationship between additive combinatorics and the geometry of monomial projective curves. We associate to a finite set of non-negative integers $A=\{a_1,\cdots, a_n\}$ a monomial projective curve $C_A\subset \mathbb P^{n-1}_{k}$ such that the Hilbert function of $C_A$ and the cardinalities of $sA:=\{a_{i_1}+\cdots+a_{i_s}\mid 1\le i_1\le \cdots \le i_s\le n\}$ agree. The singularities of $C_A$ determines the asymptotic behaviour of $|sA|$, equivalently the Hilbert polynomial of $C_A$, and the asymptotic structure of $sA$. We show that some additive in…
We undertake a systematic study of Lipschitz Normally Embedded normal complex surface germs. We prove in particular that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz Normally Embedded surface singularity.
