Del Pezzo surfaces are a foundational building block in the birational classification of algebraic varieties. For del Pezzo surfaces with “mild” (KLT) singularities, the nature of these singularities is fairly well-understood — except for a gap.
Roughly speaking:
- In characteristic zero, the possible singularities are highly constrained and fully classified
- In characteristic \(2\), there can be arbitrarily many singular points
- In characteristic \(> 3\), there can be at most four singular points
- But in characteristic \(3\), all known constructions have at most \(7\) singular points and it is not known whether this is the most possible
The problem author believes that arbitrarily many singularities are possible for characteristic \(3\) as well. The goal of this problem is to demonstrate this by construction. Note that the demonstration of a single surface with more than \(7\) singular points is just the first step. If an AI system finds such a surface, we will then ask for a new surface with more singular points than the given one, and so on. If an AI system is able to pass this iterative test ten times, we expect it will indeed have a general family of constructions that establishes the target result.
The problem author gives the AI system several general forms it may use, and believes that constructions are likely to be possible to fit into one of these forms.
Warm-up: we ask for a surface with \(7\) singularities, which is well-known and which AI systems are readily able to produce.
Attempts by AI
We have evaluated the following models on this problem. “Warm-up” refers to an easier variant of the problem with a known solution.
AI Prompts
We formulate an initial version of the problem asking simply for at least 8 singular points, but the intent is to find a general construction that can be used to generate arbitrarily many.
Warm-up
Working in characteristic 3, produce an example of a klt del Pezzo surface X with Picard number rho(X)=1 and with at least 7 singular points, constructed with one of the methods (A, B, or C) described below. Your output will be verified in Macaulay2. METHOD A — Global quotients (choose THIS if you start from a smooth CI(2,2) in P^4) Start with Y = CI(2,2) ⊂ P^4 over ZZ/3 and a tame cyclic action g (order 2) that is free in codimension 1 (so Y cap Fix(g) is 0-dimensional). Then X = Y / <g> is klt del Pezzo; singularities correspond to isolated fixed points on Y. Give your output as follows. - g as a diagonal matrix with entries either -1_S or 1_S, expressed as a list of integers each either -1 or 1 - Y as two g-invariant quadrics, using the variables x0,x1,x2,x3,x4 METHOD B — Weighted models (choose THIS if you embed X in a weighted P(w)) Realize X as a quasi-smooth weighted hypersurface or complete intersection in P(w) over ZZ/3. Choose weights prime to 3 (tame), pick degrees with positive Fano index sum(weights) - sum(degrees) > 0, and engineer the intersections with ambient singular strata (per-prime) to obtain >= 7 geometric points in total. Quasi-smoothness ensures singularities are isolated tame cyclic points counted by the strata cut. Give your output as follows. - Weights as a list of integers - One equation F (hypersurface) OR two equations F1, F2 (CI), using variables x0,x1,... as needed METHOD C — Scripted blow-ups (choose THIS if you contract HJ chains on P1xP1) Start from P1xP1 and give a basket of Hirzebruch–Jung chains, each a list of integers >= 2, describing the curves to contract.
Full problem
Working in characteristic 3, produce an example of a klt del Pezzo surface X with Picard number rho(X)=1 and with at least 8 singular points, constructed with one of the methods (A, B, or C) described below. Your output will be verified in Macaulay2. METHOD A — Global quotients (choose THIS if you start from a smooth CI(2,2) in P^4) Start with Y = CI(2,2) ⊂ P^4 over ZZ/3 and a tame cyclic action g (order 2) that is free in codimension 1 (so Y cap Fix(g) is 0-dimensional). Then X = Y / <g> is klt del Pezzo; singularities correspond to isolated fixed points on Y. Give your output as follows. - g as a diagonal matrix with entries either -1_S or 1_S, expressed as a list of integers each either -1 or 1 - Y as two g-invariant quadrics, using the variables x0,x1,x2,x3,x4 METHOD B — Weighted models (choose THIS if you embed X in a weighted P(w)) Realize X as a quasi-smooth weighted hypersurface or complete intersection in P(w) over ZZ/3. Choose weights prime to 3 (tame), pick degrees with positive Fano index sum(weights) - sum(degrees) > 0, and engineer the intersections with ambient singular strata (per-prime) to obtain >= 8 geometric points in total. Quasi-smoothness ensures singularities are isolated tame cyclic points counted by the strata cut. Give your output as follows. - Weights as a list of integers - One equation F (hypersurface) OR two equations F1, F2 (CI), using variables x0,x1,... as needed METHOD C — Scripted blow-ups (choose THIS if you contract HJ chains on P1xP1) Start from P1xP1 and give a basket of Hirzebruch–Jung chains, each a list of integers >= 2, describing the curves to contract.
Mathematician survey
The author assessed the problem as follows.
Number of mathematicians highly familiar with the problem:
a majority of those working in a subfield (≈100)
Number of mathematicians who have made a serious attempt to solve the problem:
5–10
Rough guess of how long it would take an expert human to solve the problem:
3–12 months
Notability of a solution:
a solid result in a subfield
A solution would be published:
in a strong specialty journal
Likelihood of a solution generating more interesting math:
fairly likely: the problem is rich enough that most solutions should open new avenues
Probability that the problem is solvable as stated:
60–80%