# Hodge diamonds that cannot be realised

In Paulsen–Schreieder [PS19] and vDdB–Paulsen [DBP20], the authors/we show that any block of numbers

satisfying , , and (characteristic only) can be realised as the modulo reduction of a Hodge diamond of a smooth projective variety.

While preparing for a talk on [DBP20], I came up with the following easy example of a Hodge diamond that cannot be realised integrally, while not obviously violating any of the conditions (symmetry, nonnegativity, hard Lefschetz, …).

Lemma. There is no smooth projective variety (in any characteristic) whose Hodge diamond is

Proof. If , we have , with equality for all if and only if the Hodge–de Rham spectral sequence degenerates and is torsion-free for all . Because contains an ample class, we must have equality on , hence everywhere because of how spectral sequences and universal coefficients work.

Thus, in any characteristic, we conclude that , so and the same for . Thus, is a fibration, so a fibre and a relatively ample divisor are linearly independent in the Néron–Severi group, contradicting the assumption .

Remark. In characteristic zero, the Hodge diamonds

cannot occur for any , by essentially the same argument. Indeed, the only thing left to prove is that the image cannot be a surface. If it were, then would have a global 2-form; see e.g. [Beau96, Lemma V.18].

This argument does not work in positive characteristic due to the possibility of an inseparable Albanese map. It seems to follow from Bombieri–Mumford’s classification of surfaces in positive characteristic that the above Hodge diamond does not occur in positive characteristic either, but the analysis is a little intricate.

Remark. On the other hand, the nearly identical Hodge diamond

is realised by , where is a curve of genus . This is some evidence that the full inverse Hodge problem is very difficult, and I do not expect a full classification of which Hodge diamonds are possible (even for surfaces this might be out of reach).

References.

[Beau96] A. Beauville, Complex algebraic surfaces. London Mathematical Society Student Texts 34 (1996).

[DBP20] R. van Dobben de Bruyn and M. Paulsen, The construction problem for Hodge numbers modulo an integer in positive characteristic. Forum Math. Sigma (to appear).

[PS19] M. Paulsen and S. Schreieder, The construction problem for Hodge numbers modulo an integer. Algebra Number Theory 13.10, p. 2427–2434 (2019).

# An interesting Noether–Lefschetz phenomenon

The classical Noether–Lefschetz theorem is the following:

Theorem. Let be a very general smooth surface of degree . Then the natural map is an isomorphism.

If is a smooth proper family over some base (usually of finite type over a field), then a property holds for a very general if there exists a countable intersection of nonempty Zariski opens such that holds for for all .

In general, Hilbert scheme arguments show that the locus where the Picard rank is ‘bigger than expected’ is a countable union of closed subvarieties of (the Noether–Lefschetz loci), but it could be the case that this actually happens everywhere (i.e. ). The hard part of the Noether–Lefschetz theorem is that the jumping loci are strict subvarieties of the full space of degree hypersurfaces.

If is a family of varieties over an uncountable field , then there always exists a very general member with . But over countable fields, very general elements might not exist, because it is possible that even when .

The following interesting phenomenon was brought to my attention by Daniel Bragg (if I recall correctly):

Example. Let (the algebraic closure of the field of elements, but the bar is not so visible in MathJax), let (or some scheme covering it if that makes you happier) with universal family of elliptic curves, and let be the family of product abelian surfaces . Then the locus

is exactly the set of -points (so it misses only the generic point).

Indeed, , and every elliptic curve over has . But the generic elliptic curve only has .

We see that the Noether–Lefschetz loci might cover all -points without covering , even in very natural situations.

# Local structure of finite unramified morphisms

It is well known that a finite étale morphism of schemes is étale locally given by a disjoint union of isomorphisms, i.e. there exists an étale cover such that the pullback is given by . Something similar is true for finite unramified morphisms:

Lemma. Let be a finite unramified¹ morphism of schemes. Then there exists an étale cover such that the pullback is given by , where are closed immersions of finite presentation.

Proof. Let be a point, let be the strict henselisation of at , and let be the base change of along . Then is unramified, so by Tag 04GL it splits as

where is surjective for each and no prime of lies above . But is also finite, so by Tag 00GU the map hits the maximal ideal if . Thus, we conclude that , hence is a product of quotients of .

But is the colimit of for an étale neighbourhood inducing a separable extension . Since is of finite presentation, each of the ideals and the projections are defined over some étale neighbourhood . Then the pullback is given by a finite disjoint union of closed immersions in .

Then might not be a covering, but since was arbitrary we can do this for each point separately and take a disjoint union.

Remark. The number of needed is locally bounded, but if is not quasi-compact it might be infinite. For example, we can take an infinite disjoint union of points, and such that the fibre over for has points.

Remark. In the étale case, we may actually take finite étale, by taking to be the Galois closure of , which exists in reasonable cases². For example, if is normal, we may take to be the integral closure of in the field extension corresponding to the Galois closure of . In general, if is connected it follows from Tag 0BN2 that a suitable component of the -fold fibre product of over is a Galois closure of . If the connected components of are open, apply this construction to each component.

In the unramified case, this is too much to hope for. For example, if , then we may take to be a nontrivial finite étale cover of an elliptic curve . This is finite and unramified, but does not split over any finite étale cover of since there aren’t any. In fact, it cannot split over any connected étale cover whose image contains , since that implies the image only misses finitely many points (as is ample), which is again impossible since .

¹For the purposes of this post, unramified means in the sense of Grothendieck, i.e. including the finite presentation hypothesis. In Raynaud’s work on henselisations, this was weakened to finite type. See Tag 00US for definitions.

²I’m not sure what happens in general.

# Rings that are localisations of each other

This is a post about an answer I gave on MathOverflow in 2016. Most people who have ever clicked on my profile will probably have seen it.

Question. If and are rings that are localisations of each other, are they necessarily isomorphic?

In other words, does the category of rings whose morphisms are localisations form a partial order?

In my previous post, I explained why and are not isomorphic, even as rings. With this example in mind, it’s tempting to try the following:

Example. Let be a field, and let . Let

be an infinite-dimensional polynomial ring over , and let

Then is a localisation of , and we can localise further to obtain the ring

isomorphic to by shifting all the indices by 1. To see that and are not isomorphic as rings, note that is closed under addition, and the same is not true in .

Is there a moral to this story? Not sure. Maybe the lesson is to do mathematics your own stupid way, because the weird arguments you come up with yourself may help you solve other problems in the future. The process is more important than the outcome.

# Is the affine line isomorphic to the punctured affine line?

This is the story of Johan Commelin and myself working through the first sections of Hartshorne almost 10 years ago (nothing creates a bond like reading Hartshorne together…). This post is about problem I.1.1(b), which is essentially the following:

Exercise. Let be a field. Show that and are not isomorphic.

In my next post, I will explain why I’m coming back to exactly this problem. There are many ways to solve it, for example:

Solution 1. The -algebra represents the forgetful functor , whereas represents the unit group functor . These functors are not isomorphic, for example because the inclusion induces an isomorphism on unit groups, but not on additive groups.

A less fancy way to say the same thing is that all -algebra maps factor through , while the same evidently does not hold for -algebra maps .

However, we didn’t like this because it only shows that and are not isomorphic as -algebras (rather than as rings). Literal as we were (because we’re undergraduates? Lenstra’s influence?), we thought that this does not answer the question. After finishing all unstarred problems from section I.1 and a few days of being unhappy about this particular problem, we finally came up with:

Solution 2. The set is closed under addition, whereas is not.

This shows more generally that and are never isomorphic as rings for any fields and .

# P¹ is simply connected

This is a cute proof that I ran into of the simple connectedness of . It does not use Riemann–Hurwitz or differentials, and instead relies on a purely geometric argument.

Lemma. Let be an algebraically closed field. Then is simply connected.

Proof. Let be a finite étale Galois cover with Galois group . We have to show that is an isomorphism. The diagonal is ample, so the same goes for the pullback to [Hart, Exc. III.5.7(d)]. In particular, is connected [Hart, Cor. III.7.9].

But is isomorphic to copies of because the action

is an isomorphism. If is connected, this forces , so is an isomorphism.

The proof actually shows that if is a smooth projective variety such that is a set-theoretic complete intersection of ample divisors, then is simply connected.

Example. For a smooth projective curve of genus , the diagonal cannot be ample, as . We already knew this by computing the self-intersection , but the argument above is more elementary.

References.

[Hart] Hartshorne, Algebraic geometry. GTM 52, Springer, 1977.

# Number theory is heavy metal

As some of you may be aware, I am a musician as well as a mathematician. I often like to compare my experiences between the two. For example, I found that my approach to the creative process is not dissimilar (in both, I work on the more technical side, with a particular interest in the larger structure), and I face the same problems of excessive perfectionism (never able to finish anything).

This post is about some observations about the material itself, as opposed to my interaction with it.

Universal donor.

A look at the arXiv listing for algebraic geometry shows the breadth of the subject. Ideas from algebraic geometry find their way into many different areas of mathematics; from representation theory and abstract algebra to combinatorics and from number theory to mathematical physics. But when I say algebraic geometry is a universal donor (of ideas, techniques, etc.), I also mean that its applications to other fields far outnumber the applications of other fields to algebraic geometry, and that the field of algebraic geometry is largely self-contained.

Much the same role is played by classical music inside musical composition. Common practise theory is used throughout Western music, whether you’re listening to hip hop, trance, blues, ambient electronic, bluegrass, or hard rock. Conversely, the influence of popular genres music on [contemporary] classical is comparatively little. One could therefore argue that classical music is a universal donor in the field of musical composition.

Universal receptor.

The opposite is the case for number theory. Another vast area, number theory often uses ingenious arguments combining ideas from algebra, combinatorics, analysis, geometry, and many other areas. In practical terms, the amount of material that a number theorist needs to master is immense: whatever solves your particular problem.

So is there any musical genre that plays a similar role? I claim that metal fits the bill. With a vast list of subgenres including thrash metal, black metal, doom metal, progressive metal, death metal, symphonic metal, nu metal, grindcore, hair metal, power metal, and deathcore, metal writing often contains creative combinations of other genres, from classical music to ambient electronic, and from free jazz to hip hop. As Steven Wilson of Porcupine Tree said about his rediscovery of metal:

I said to myself, this is where all the interesting musicians are working! Because for a long time I couldn’t find where all these creative musicians were going… You know in the 70’s they had a lot of creative musicians like Carlos Santana, Jimmy Page, Frank Zappa, Neil Young, I was thinking “where are all these people now?” and I found them, they were working in extreme metal.

I sometimes think of metal as a small microcosmos reflecting the full range of Western (and some non-Western) music, tied together by distorted guitars and fast, technical drumming.

In summary, algebraic geometry is classical music, and number theory is heavy metal.

# Not every open immersion is an open immersion

An immersion (or locally closed immersion) of schemes is a morphism that can be factored as , where is a closed immersion and is an open immersion. If it is moreover an open morphism, it need not be an open immersion:

Example. Let be a nonreduced scheme, and let be the reduction. This is a closed immersion, whose underlying set is the entire space. Thus, it is a homeomorphism, hence an open morphism. It is not an open immersion, for that would force it to be an isomorphism.

Remark. However, every closed immersion is a closed immersion; see Tag 01IQ.

# Finiteness is not a local property

In this post, we consider the following question:

Question. Let be a Noetherian ring, and and -module. If is a finite -module for all primes , is finite?

That is, is finiteness a local property?

For the statement where local means the property is true on a cover by Zariski opens, see Tag 01XZ. Some properties (e.g. flatness) can also be checked at the level of local rings; however, we show that this is not true for finiteness.

Example 1. Let , and let . Then , because localisation commutes with direct sums and if is prime. Thus, is finitely generated for all primes . Finally, , because is torsion. But is obviously not finitely generated.

Example 2. Again, let , and let be the subgroup of fractions with such that is squarefree. This is a subgroup because can be written with denominator , and that number is squarefree if and are. Clearly is not finitely generated, because the denominators can be arbitrarily large. But , which is finitely generated over . If is a prime, then is the submodule , which is finitely generated over .

Another way to write is .

Remark. The second example shows that over a PID, the property that is free of rank can not be checked at the stalks. Of course it can be if is finitely generated, for then is finite projective [Tag 00NX] of rank , hence free since is a PID.

# Higher pushforwards along finite morphisms

This post is about one of my favourite answers I have given on MathOverflow, although it seems to have gone by mostly unnoticed. In the post, Qixiao asks (essentially) the following:

Question. If is a finite morphism of schemes, is the pushforward exact?

Note that this is true on the subcategory of quasicoherent sheaves because affine morphisms have no quasicoherent higher pushforwards. Also, in the étale topology the pushforward along a finite morphism is exact on the category of all abelian sheaves; see e.g. Tag 03QP.

However, we show that the answer to the question above is negative.

Example. Let be the spectrum of a DVR , let be a finite extension of domains such that has exactly two primes above , and let . For example, and , or and if you prefer a more geometric example.

By my previous post, the global sections functor is exact. If the same were true for , then the global sections functor on would be exact as well. Thus, it suffices to prove that this is not the case, i.e. to produce a surjection of sheaves on such that the map on global sections is not surjective.

The topological space of consists of closed points and a generic point . Let and ; then is open and is closed. Hence, for any sheaf on , we have a short exact sequence (see e.g. Tag 02UT)

where and are the inclusions. Let be the constant sheaf ; then the same goes for and . Then the map

is given by the diagonal map , since is connected by has two connected components. This is visibly not surjective.