On RC varieties without smooth rational curves
Ilya Karzhemanov

TL;DR
This paper constructs high-dimensional rationally connected varieties that notably lack any smooth rational curves, challenging previous assumptions about their geometric properties.
Contribution
It introduces the first examples of rationally connected varieties without smooth rational curves, expanding understanding of their possible structures.
Findings
Existence of high-dimensional rationally connected varieties without smooth rational curves
Construction methods for such varieties
Implications for the geometry of rationally connected varieties
Abstract
We construct normal rationally connected varieties (of arbitrarily large dimension) not containing any smooth rational curves.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
On RC varieties without smooth rational curves
Ilya Karzhemanov
Abstract.
We construct normal rationally connected varieties (of arbitrarily large dimension) not containing any smooth rational curves.
MS 2010 classification: 14M22, 14H45, 14H60
Key words: rationally connected variety, moduli of vector bundles
1. Introduction
1.1.
Let be a rationally connected projective variety, not necessarily smooth or normal, defined over . Then any smooth is covered by various images of morphisms such that the pullback is ample (see e.g. [11, Theorem 3.7]). One also observes that the image is smooth in this case for a generic choice of (see [11, Theorem 3.14]).
The present paper grew out of an attempt to understand whether the preceding property holds for an arbitrary as well (see [9], [10] for some related results). Namely, if is normal and rationally connected, is there always at least one smooth rational curve on ? Or more generally, i.e. dropping the normality assumption, are there such rationally connected that don’t contain any smooth rational curves, with being arbitrarily large?
In the non-normal case, the answer is evident when (normalization), although even for the product we are not sure how to proceed (there might exist with degree projection onto ). So the assumption makes the problem more interesting (we refer to Section 4 for further variations).
Our main result treats the normal case as follows:
Theorem 1.2**.**
There exists normal rationally connected variety without smooth rational curves (we call such weird) and with arbitrarily large.
Let us briefly outline the strategy of the proof of Theorem 1.2.
1.3.
To make the construction easier we’d like to consider those that parameterize certain geometric objects. This should, in principle, allow one interpret smoothness of any rational curve in terms of properties of the corresponding family of objects. We’ve given our preference to the moduli spaces of rank and vector bundles over an algebraic curve of genus (compare with [5] for another usage of in birational geometry).
Recall that is a Fano variety (see Section 2 for its specific properties). Yet, unfortunately, it is too early to just set and conclude the proof of Theorem 1.2:
Example 1.4*.*
Given any stable vector bundle , after twisting by for some (independent of ) one may identify with an extension class from . This provides a rational dominant map (see [7, Proposition 7.9] for instance) and shows that is in fact unirational. Now comes the discouraging observation that actually contains plenty of smooth rational curves. Namely, starting with any point , some generic vector bundle with determinant , and a linear form on the fiber , we consider the morphism of sheaves corresponding to . The kernel of this morphism is again a rank vector bundle; it is stable due to [12, Lemma 5.2] and has trivial determinant. This shows that the whole embeds into (cf. [13, 5.9]).
Recall that all the (equivalent) constructions of involve certain GIT quotient of a source space by some group . This can be the space of relative Grassmannians over (resp. ), the space of -representations of (resp. ), the space of flat connections on a fixed rank topologically trivial vector bundle over (resp. the gauge group), etc. Our idea then was, starting with a smooth rational curve , lift it in a manner to and obtain a contradiction with the fact that is affine or something like this.
However, as Example 1.4 shows, this idea won’t work directly. The reason behind is that the claimed “lifting to ” doesn’t exist: there’s always an ambiguity in the choice of a point to associate with any given point on . In order to circumvent this we use another construction of (compare with [15, 5.1]). Namely, after some care (see Section 2), one may take the group of special -matrices with coefficients in a formal power series ring (resp. some ind-group). Then for a particular locus (see 3.4), with any point on one associates (canonically) a point in , modulo some moderate assumptions (cf. Proposition 3.2).
Remark 1.5*.*
This seems to be another natural object, in addition to the theta-divisor (see 2.5 below), which comes for free with . It would be interesting to explore the relation between and further: whether, say, rationality (resp. precise dimension) statement for (see e.g. [15, 3.1]) holds also for ?
One observes next that is normal, projective and rationally connected – the facts we derive from the ind-construction of . This readily shows that there is no smooth rational , since otherwise it’ll be lifted to direct limit of affine varieties, which is impossible (see Section 3 for details).
2. Preliminaries
2.1.
Let the notation be as in 1.3. We first briefly recall the ind-construction of the moduli spaces (see [1], [2], [8] for a complete account). For this we fix a point , a small disk around and a local coordinate on such that . We also put and .
Let be a vector bundle on of rank and . One proves by induction on that is trivial over (cf. [1, Lemma 3.5]). Then trivializing also over provides an element whose class in the space uniquely determines up to the left action on of the subgroup (i.e. up to a choice of trivialization for E\big{|}_{X^{*}}).
Equivalently, one may associate with a special lattice (with ) generated as a -module by global sections of E\big{|}_{X^{*}} and such that (or ) is reconstructed from (see [1, Proposition 2.3]). In particular, one regards as a collection of all such (with the obvious -action) and represents it as a direct limit of irreducible projective varieties , which consist of those that have a basis with , and for all (see [1, Theorem 2.5, Proposition 2.6]).
We will need the following simple observation:
Lemma 2.2** (cf. [1, Lemma 4.5]).**
With notation as above, every -orbit of is represented by some matrix in (by definition is the image of under the quotient morphism ), so that admits a basis consisting of vectors from . In particular, one may choose to satisfy , where is the identity matrix.
2.3.
Further, recall that there’s a canonical isomorphism in codimension , where (resp. ) is considered as an algebraic (resp. quotient) stack (see [1, Proposition 3.4, Lemma 8.2] and Remark 2.4 below). In addition, the group also carries the structure of an integral ind-scheme, so that its action on is compatible with the underlying ind-structure (cf. [1, Proposition 6.4]).
Altogether this yields an ind-structure on , so that as an open ind-subscheme with codimension complement, and an ample line bundle \mathcal{L}\big{|}_{\mathcal{SU}_{X}(r)} with the pullback under the natural projection being some -invariant (determinant) line bundle (see [1, Sections 5, 7, 8]). The global sections from (the space of conformal blocks), , coincide by construction with the -invariant global sections of (see [1, Theorem 8.5]). We’ll mention a few more properties of in 2.5.
Remark 2.4*.*
Using the (mappings by) global sections of , one may regard both and as essentially the same scheme with two complementary structures: of a quasi-projective variety and an ind-scheme. In particular, any open (resp. closed) ind-subscheme of corresponds, tautologically, to an open (resp. closed) subscheme in , and vice versa (compare with the proof of Theorem 7.7 in [1]).
2.5.
To conclude this section, let’s fix a bit more of notation/conventions and recall some auxiliary facts, as those will be used later in Section 3.
Choose some semi-stable bundle (identified with a point in the corresponding moduli space) with a Hermitian structure specifying the gauge group action on the space of all connections on . Using [15, 3.2] (cf. [7, Lemma 4.2]) and the main result of [6] (cf. [14]) one represents as a sum of stable bundles.
Note that any two direct sum decompositions differ only by the order of summands. Indeed, otherwise arguing by induction on one obtains a surjection between two stable bundles of degree [math], which is impossible. This leads to the following:
Theorem 2.6**.**
There exists a unique (up to ) flat unitary connection on .
Finally, although we’ll not quite need this, let us mention for consistency some other facts about (see also Section 4). First of all, variety is locally factorial, with Picard group generated by (see [7, Theorems A, B]). One may equivalently interpret as the so-called theta-divisor. The latter consists of all for which with respect to some fixed cycle on of degree . Furthermore, the canonical class equals , so that is a Fano variety.
3. Proof of Theorem 1.2
3.1.
We retain the notation of Section 2. Put and let (resp. ) be as in Lemma 2.2. Recall that E\big{|}_{\Delta}=\mathbb{C}^{r}\times\Delta with constant basis, while the -module E\big{|}_{X^{*}} is generated by , so that the two data are glued over via . We will additionally assume that .
To apply the strategy outlined in 1.3 one should have (at least) the following:
Proposition 3.2**.**
In the previous setting, carries a unique (Bohr-Sommerfeld) basis, which does not depend on the choice of . In fact, there is only one associated with (cf. 2.1), satisfying .
Proof.
According to our assumption there exists a collection of vectors that generates E\big{|}_{X^{*}} and coincides with the standard basis of modulo . Using this, the flat unitary connection on (see Theorem 2.6), plus the preceding description of in terms of and , we construct (via the parallel transport starting with ) global -sections of , which generate E\big{|}_{X^{*}} and satisfy the equation in a small disk around .
More precisely, since is flat and ,1)1)1)Here is the usual Kählher differential of the ring . equation and its solutions do not depend on the choice of (aka ). In particular, there is a correctly defined (canonical) extension of every over the entire , as claimed. We also observe that by uniqueness of the solutions all entries in are some elements from because locally on . This gives the desired basis for .
Recall next that is unique up to . Now the last claim of proposition follows from the fact that one may take to be the columns of and that for any by definition. ∎
Remark 3.3*.*
(Notation as in the proof of Proposition 3.2) The sections provide an isomorphism of -bundles.2)2)2)The bundle carries a natural Hermitian structure which varies together with . Let be the moduli space of all completely reducible flat connections on . Then conversely, for any -trivial the corresponding can be chosen to satisfy . Indeed, any such is generated by -flat -sections , obtained from some constant sections of via fiberwise (gauge) transformation. Locally on one has modulo by construction, which implies that . Then all depend only on due to and uniqueness of the solutions. Note that this construction of requires to be fixed. It also shows that any other lattice , for , has the corresponding equal to as well. Indeed, recall that the condition for to be -flat translates into modulo , which must be gauge invariant.
3.4.
The locus from Remark 3.3 is our candidate for the weird rationally connected variety . Let’s study the geometry of more closely by employing its description in terms of the lattices . But first we make the following:
Assumption*.*
Fix and choose the initial curve to be generic of genus . This gives for generic (take for instance the direct sum of different line bundles satisfying and for all ).
We will write in what follows (cf. 2.3 and Remark 2.4). This aims to just simplify the notation and won’t cause any loss of generality.
Consider the subset of all lattices satisfying as in 3.1. Let also be the locus of all matrices for various . It follows from the previous constructions that maps onto under the quotient morphism .
Lemma 3.5**.**
* is a -invariant closed ind-subscheme.*
Proof.
Consider some . Recall that uniquely determines the corresponding (see Proposition 3.2). Then, since the group acts freely on , the set may be identified with near .
In particular, any (bounded in analytic topology) Cauchy sequence of lattices yields the corresponding sequence of , as follows from the definition of . One then (obviously) has a limit that maps to . This shows that is closed in analytic topology. Now using the representation from 2.3 we obtain that is a closed ind-subscheme.
Finally, the -invariance of follows from Remark 3.3, which concludes the proof. ∎
From Lemma 3.5 we obtain
[TABLE]
for the natural -equivariant morphism of ind-schemes.
Proposition 3.6**.**
* is a normal, projective and rationally connected variety.*
Proof.
It follows from the above Assumption that the group acts freely at the general point on . This gives
Lemma 3.7**.**
* is irreducible and reduced (i.e. integral).*
Proof.
Recall that and are isomorphic at the general point (cf. the proof of Lemma 3.5). Note also that is integral (compare with the proof of [1, Proposition 2.6]). Hence is integral as well. Then, since (equivariantly) at the general point, the claim follows from [1, Proposition 6.4, Lemma 6.3, d)]. ∎
From the definition of we deduce that the locus is closed in analytic topology (cf. Lemma 3.5 and Remark 2.4). Hence is a projective integral variety.
Further, consider some lattice , identified with the matrix as in the proof of Lemma 3.5. Choose another lattice , with associated matrix , and define the curve
[TABLE]
Lemma 3.8**.**
* is a rational curve.*
Proof.
Indeed, there’s an obvious birational map of ind-schemes (for tautologically), which yields a rational dominant map . ∎
It follows from Lemmas 3.7 and 3.8 that is rationally connected (cf. [11, Ch. IV, Proposition 3.6]). Thus it remains to prove normality.
Firstly, the proof of [1, Proposition 6.1] shows that is smooth in codimension , which implies that is normal because it is a complete intersection on (more precisely, is a direct limit of finite-dimensional complete intersections, which are smooth in codimension , hence normal by Serre’s criterion). Then is also normal for acting freely on .
Secondly, since for any lattice , the condition for some is of codimension on . Thus we get and is normal – all in codimension .
Finally, if is the ample generator of (see 2.5), it follows from the quotient construction of that any function in can be represented as a ratio of global sections of various restrictions \mathcal{L}^{c}\big{|}_{\mathcal{D}},c\gg 1. In particular, the property is satisfied for all the rational functions on , and so is normal by Serre’s criterion.
Proposition 3.6 is completely proved. ∎
From (the proof of) Proposition 3.6 we deduce
Corollary 3.9**.**
* has codimension (hence in particular can be made arbitrarily large).*
Proof.
This follows from the fact that the locus is defined by the equation for generic matrix (cf. 3.4 and Lemma 2.2). ∎
3.10.
Now let’s turn to the proof of Theorem 1.2. Suppose that is a smooth rational curve. Associating with any point the lattice as in 2.1 yields an affine bundle over . More specifically, since every carries the canonical basis of Proposition 3.2, using it one identifies with the affine space .
Further, since and the bundle is locally analytically trivial by construction, we get for some . Note that for . Then taking a global section of gives an embedding .
Thus we obtain a non-constant family of matrices algebraically parameterized by . In particular, there exists a rational function such that for all , which is absurd. The proof of Theorem 1.2 is complete (cf. Proposition 3.6 and Corollary 3.9).
4. Some questions and comments
We’d like to conclude the paper by asking the following questions:
- •
Is there a weird over a field of positive characteristic (cf. [3], [4])? Similarly, modifying the notion of rational connectivity accordingly, is there a non-Kähler compact complex (maybe even smooth) weird (cf. [16])?
- •
Is the locus a Fano variety (and how to express it in terms of theta-divisors)? Does it have locally factorial singularities (resp. what is its Picard group)? Same questions for any weird .
- •
By applying the weak factorization theorem it would be interesting to find out whether being weird provides an obstruction for variety to be rational. What about the case of again (cf. Remark 1.5)?
- •
Note that the locus consists entirely of strictly semi-stable bundles (cf. Remark 3.3). In particular, when , writing any as a sum of two line bundles one finds that there is a -cover (i.e. is the Kummer variety).3)3)3)This example shows that the Assumption made in 3.4 is crucial for to be rationally connected. Indeed, for the locus is not rationally connected (as ), and the reason is that one lacks the canonical correspondence here (cf. the proof of Lemmas 3.7 and 3.8). Is the same true for an arbitrary (with replaced by an appropriate Abelian variety and “” by “generically Galois”)? Similar question for any weird .
- •
Is it possible to find weird in any given dimension (cf. Corollary 3.9)?
Acknowledgments. I’d like to thank A. Beauville, S. Galkin, J. Kollár, and T. Milanov for their interest, valuable comments and references. Most of the paper was written during my visits to MIT (US), UOttawa (Canada) and PUC (Chile) in April-May 2015. I am grateful to these Institutions and people there for hospitality. The work was supported by World Premier International Research Initiative (WPI), MEXT, Japan, and Grant-in-Aid for Scientific Research (26887009) from Japan Mathematical Society (Kakenhi).
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] A. Beauville and Y. Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), no. 2, 385 – 419.
- 2[2] A. Beauville, Y. Laszlo and C. Sorger, The Picard group of the moduli of G 𝐺 G -bundles on a curve, Compositio Math. 112 (1998), no. 2, 183 – 216.
- 3[3] F. Bogomolov and Y. Tschinkel, Rational curves and points on K 3 𝐾 3 K 3 surfaces, Amer. J. Math. 127 (2005), no. 4, 825 – 835.
- 4[4] F. Bogomolov, B. Hassett and Y. Tschinkel, Constructing rational curves on K 3 surfaces, Duke Math. J. 157 (2011), no. 3, 535 – 550.
- 5[5] A.-M. Castravet, Examples of Fano varieties of index one that are not birationally rigid, Proc. Amer. Math. Soc. 135 (2007), no. 12, 3783 – 3788 (electronic).
- 6[6] S. K. Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. 18 (1983), no. 2, 269 – 277.
- 7[7] J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), no. 1, 53 – 94.
- 8[8] G. Faltings, A proof for the Verlinde formula, J. Algebraic Geom. 3 (1994), no. 2, 347 – 374.
