In a previous post I introduced the notion of (quasi-)phantom categories, and discussed Kuznetsov's conjecture stating that they should not exist. This post has lain dormant for too long, but I decided to finally finish it this evening. The goal is clear: indicate how we can construct phantom categories, tell how results in classical algebraic geometry help us in doing so and explain what the abstract properties of some of the (quasi)phantom categories indicate about the surface.

## The counterexamples

They are in roughly chronological order, based on the date of the arXiv preprints. The easiest case to understand is probably the Beauville surface, which is not the first example that was obtained (at least in the form of a publicly available preprint).

### Classical Godeaux surface

This surface is a $\mathrm{Cyc}_5$-quotient of the Fermat quintic surface in $\mathbb{P}^3$, i.e. we take the fifth roots of unity acting on the four coordinates. The explicit construction as a quotient allows for strong structural results, a phenomenon we will see again for the Beauville surface.

In this case, the length of the exceptional collection is 11, which is the longest such a sequence can get. The complement in the Grothendieck group is (by the quotient construction) $\mathrm{Cyc}_5$. The complement in the Hochschild homology on the other hand is zero, which is a Hochschild-Kostant-Rosenberg computation.

### Burniat surfaces

A first class of quasi-phantoms is obtained by considering the Burniat surfaces of degree 6 (i.e. $\mathrm{K}_X^2=6$). This is done in Derived categories of Burniat surfaces and exceptional collections. This type of Burniat surfaces can be described as Galois covers (with group $\mathrm{Cyc}_2^2$, i.e. 4-fold covers) of a del Pezzo surface of degree 6, which are the blow-ups of $\mathbb{P}^2$ in 3 points not on a line. We have strong structural results on the derived category of del Pezzo's, and these can be lifted to the Burniat surfaces of degree 6.

The idea is to use the 3-block exceptional sequence of length 6 on such a del Pezzo. One can then prove that the lift of this sequence to the cover is still an exceptional collection, but now with some nonzero $\mathrm{Ext}^2$'s. This tells us that the endomorphism algebra, which describes the triangulated category generated by the exceptional collection, is formal, and independent of the Burniat surface under consideration (there is a 4-dimensional family of these). This has the funny consequence that the quasi-phantom category (which was conjectured to not exist) actually contains all the non-trivial information on the derived level!

The fact that the Grothendieck group of the orthogonal complement is $\mathrm{K}_0(\mathcal{A})$ is $\mathrm{Cyc}_2^6$ is a classical result, which is related to the degree of the surface as explained in section 3 of On certain examples of surfaces with $\mathrm{p}_g=0$ due to Burniat.

### Determinantal Barlow surfaces

These surfaces (constructed using a $\mathrm{Dih}_{10}$-action on $\mathbb{P}^3$) also admit an exceptional sequence of length 11. And what is more interesting about these surfaces is that their Grothendieck group is free of rank 11. Hence the orthogonal complement of the sequence of length 11 is a true phantom category!

### Beauville surfaces

This is a class of surfaces obtained by taking a suitable quotient of the product of two curves. The example considered in Exceptional collections of line bundles on the Beauville surface is of a $\mathrm{Cyc}_5\times\mathrm{Cyc}_5$-action on the Fermat quintic curve $X^5+Y^5+Z^5=0$ (which is of genus 6). Because of the construction as a quotient it is again possible to obtain strong structural results for these surfaces.

The interesting property of Beauville surfaces is that they can be considered as fake quadrics: they look and behave like $\mathbb{P}^1\times\mathbb{P}^1$ in many respects. But now we take non-trivial curves, and incorporate a group action. One of the similarities is that they have an exceptional collection of line bundles of length 4. But unlike the quadric surface case this collection is not full: its orthogonal complement will be a quasi-phantom category, whose Grothendieck group is the torsion part of the Picard group, which is $\mathrm{Cyc}_5\times\mathrm{Cyc}_5$.

### Fake projective planes

We have had fake quadrics, but there are also fake projective planes, which are surfaces that share numerical invariants with the projective plane, yet are of general type.

### Fake del Pezzo surfaces

When the first examples had been obtained, the question became whether it is possible to unify these examples and see why they are (quasi)phantoms. This has been done in Enumerating exceptional collections on some surfaces of general type with $\mathrm{p}_{g} = 0$. What is so interesting about this preprint is that the code is publicly available.

## Summary

To summarise the results we give the following table. We obtain the (quasi-)phantom category $\mathcal{A}$ as the orthogonal complement of an exceptional collection, whose length gives us the rank of the freely generated part of the Grothendieck group.

type $\mathrm{K}_0(\mathcal{A})$ length of the exceptional collection quasi-phantom $\mathrm{Cyc}_5$ 11 quasi-phantom $\mathrm{Cyc}_2^6$ 6 phantom 0 11 quasi-phantom $\mathrm{Cyc}_5^2$ 4

This table is incomplete, but the number of known quasiphantom categories increases rapidly, hence only the (as far as I can tell) first four examples are listed. Maybe I will elaborate further on the subject of (quasi)phantoms later, right now I just want to get this post finished.