[Paper Review] On an analogue of the Markov equation for exceptional collections of length 4

Louis de Thanhoffer de Völcsey, Michel Van den Bergh|arXiv (Cornell University)|Jul 14, 2016

Geometric Analysis and Curvature Flows5 references6 citations

TL;DR

This paper classifies solutions to a system of Diophantine equations derived from numerical constraints on full exceptional collections of length 4 on surfaces, generalizing the classical Markov equation from rank 3 to rank 4. It shows that all solutions are either geometric (corresponding to $\mathbb{P}^2$, $\mathbb{P}^1\times\mathbb{P}^1$, or blowups of $\mathbb{P}^2$) or arise from a 'numerical blowup' procedure, with infinitely many non-geometric solutions realizable as noncommutative surfaces.

ABSTRACT

We classify the solutions to a system of equations, introduced by Bondal, which encode numerical constraints on full exceptional collections of length 4 on surfaces. The corresponding result for length 3 is well-known and states that there is essentially one solution, namely the one corresponding to the standard exceptional collection on the surface $\mathbb{P}^2$. This was essentially proven by Markov in 1879. It turns out that in the length 4 case, there is one special solution which corresponds to $\mathbb{P}^1 imes\mathbb{P}^1$ whereas the other solutions are obtained from $\mathbb{P}^2$ by a procedure we call numerical blowup. Among these solutions, three are of geometric origin ($\mathbb{P}^2\cup \{\bullet\}$, $\mathbb{P}^1 imes\mathbb{P}^1$ and the ordinary blowup of $\mathbb{P}^2$ at a point). The other solutions are parametrized by $\mathbb{N}$ and very likely do not correspond to commutative surfaces. However they can be realized as noncommutative surfaces, as was recently shown by Dennis Presotto and the first author.

Motivation & Objective

• To classify solutions to a system of Diophantine equations encoding numerical constraints on full exceptional collections of length 4 on surfaces, as introduced by Bondal.
• To extend the classical Markov equation classification from rank 3 (corresponding to $\mathbb{P}^2$) to rank 4, identifying new geometric and non-geometric solutions.
• To determine which solutions correspond to actual smooth projective surfaces and which require noncommutative geometric realizations.
• To establish that all solutions are related by signed braid group actions, and to identify canonical representatives under this equivalence.

Proposed method

• The authors analyze the Gram matrix of the Euler form for a full exceptional collection of length 4, constrained by unipotency of the Serre functor on the numerical Grothendieck group.
• They derive a system of two Diophantine equations (1.3) from the unipotency condition on the Serre automorphism in rank 4.
• The classification proceeds by analyzing minimal solutions under the action of the signed braid group, using mutation and rotation operations.
• The method involves case analysis based on inner products $\langle e_1,e_2 \rangle$ and $\langle e_3,e_4 \rangle$, reducing to subcases where these are 2 or less.
• The authors use the concept of 'numerical blowup' to construct new solutions from known ones, generalizing the construction from rank 3.
• They verify that solutions are inequivalent via invariants such as $\delta(K) = 9 - n^2$ and mod 2 behavior of the Serre automorphism.

Experimental results

Research questions

• RQ1 What are all the solutions to the system of Diophantine equations that encode numerical constraints on full exceptional collections of length 4 on surfaces?
• RQ2 Which of these solutions correspond to actual smooth projective surfaces, and which require noncommutative geometric realizations?
• RQ3 How are the solutions related under the action of the signed braid group, and can they be classified up to equivalence?
• RQ4 What is the geometric or noncommutative origin of the solutions in the infinite family parameterized by $n \in \mathbb{N}$?
• RQ5 Is there a canonical representative for each orbit of solutions under braid group action, and how can they be distinguished?

Key findings

• The only geometric solutions are $\mathbb{P}^2$, $\mathbb{P}^1\times\mathbb{P}^1$, and the first Hirzebruch surface $\mathbb{F}_1$, corresponding to $n=0$, $n=1$, and the special solution $M_{\text{P}^1\times\text{P}^1}$.
• All solutions are equivalent under the signed braid group to one of two canonical forms: the $\mathbb{P}^1\times\mathbb{P}^1$ solution or the infinite family $K_n$ for $n \in \mathbb{N}$.
• The solution corresponding to $\mathbb{P}^1\times\mathbb{P}^1$ has $\delta(K) = 8$, while the family $K_n$ has $\delta(K) = 9 - n^2$, distinguishing them from $K_1$.
• The solution $M = \begin{bmatrix}1&2&2&4\\ 0&1&0&2\\ 0&0&1&2\\ 0&0&0&1\end{bmatrix}$ corresponds to the standard exceptional collection on $\mathbb{P}^1\times\mathbb{P}^1$, with $\delta(K) = 8$.
• The infinite family $K_n$ for $n \in \mathbb{N}$ includes $n=0$ (corresponding to $\mathbb{P}^2$) and $n=1$ (corresponding to $\mathbb{F}_1$), with higher $n$ yielding non-geometric solutions.
• Solutions with $n \geq 2$ are not realizable on any smooth projective surface but are realized as noncommutative surfaces, as shown by Presotto and the first author in [dTdVP].