Linked determinantal loci and limit linear series
We study (a generalization of) the notion of linked determinantal loci recently introduced by the second author, showing that as with classical determinantal loci, they are Cohen-Macaulay whenever they have the expected codimension. We apply this to prove Cohen-Macaulayness and flatness for moduli spaces of limit linear series, and to prove a comparison result between the scheme structures of Eisenbud-Harris limit linear series and the spaces of limit linear series recently constructed by the second author. This comparison result is crucial in order to study the geometry of Brill-Noether loci via degenerations.
The theory of limit linear series for curves of compact type was developed by Eisenbud and Harris in a series of papers in the 1980’s, with the foundational definitions and results appearing in [EH86]. They were able to give spectacular applications (see for instance [EH87a] and [EH87b]), despite the fact that the moduli space of limit linear series they constructed for families of curves was not proper. However, for finer analyses, it becomes important to have a proper moduli space. This arises for instance when one wants to carry out intersection theory calculations on moduli spaces of linear series, as in Khosla [Kho], or when one wants to study the geometry of moduli spaces of linear series, as in the current work of Castorena-Lopez-Teixidor [CLT] and Chan-Lopez-Pflueger-Teixidor [CLPT]. A major step in this direction was accomplished in [Oss14b], when an equivalent definition of limit linear series was introduced, leading to the first proper moduli spaces in families. However, while the new definition was shown to agree with the Eisenbud-Harris definition on a set-theoretic level, the scheme structures are difficult to compare directly. The Eisenbud-Harris scheme structure is more amenable to explicit calculation, but without knowing that the two scheme structures agree, one cannot carry through the arguments of [CLT] and [CLPT], because different non-reduced structures in the special fiber will typically affect more elementary aspects of the generic fiber, such as connectedness or the genus.
In the present paper, we address this issue by showing that under the typical circumstances considered in limit linear series arguments, the two scheme structures do in fact agree. We further show that when they have the expected dimension, limit linear series spaces are Cohen-Macaulay and flat. Our arguments center around an analysis of the “linked determinantal loci” introduced in Appendix A of [Oss14a] in order to prove smoothing theorems for limit linear series. In fact, we study a more general definition than the one considered in [Oss14a], which is also a generalization of classical determinantal loci. A preliminary definition is the following.
Let be a scheme, and be positive integers. Suppose that are vector bundles of rank on and we have morphisms
for each . Given , we say that is an -linked chain if the following conditions are satisfied:
For each ,
On the fibers of the at any point with , we have that for each ,
On the fibers of the at any point with , we have that for each ,
We then define linked determinantal loci as follows.
Let be an -linked chain on a scheme . Given , suppose are vector bundles of rank and respectively, and let and be any morphisms. Then the associated linked determinantal locus is the closed subscheme of on which the induced morphisms
have rank less than or equal to for all .
Note that the case and recovers the usual notion of determinantal locus. It appears a priori that the codimension could be much larger than for a single determinantal locus, but it turns out that the different determinantal conditions we are imposing are highly dependent. The case considered in Appendix A of [Oss14a] amounted to requiring that and be quotient maps onto bundles of rank . In that situation, it was shown that the classical codimension bound also applies to linked determinantal loci. However, Cohen-Macaulayness does not appear to be addressable by the methods used in loc. cit. Our main theorem is thus the following:
If is Noetherian, then in the situation of Definition 1.2, every irreducible component of the linked determinantal locus has codimension at most . Moreover, if equality holds, and is Cohen-Macaulay then the linked determinantal locus is also Cohen-Macaulay.
The proof proceeds by considering a suitable universal version of the linked determinantal locus, and showing in essence that it is a “partial initial degeneration” of the classical universal determinant locus. That is to say, in terms of ideals of minors we show that our locus is defined by zeroing out monomials in each minor according to a certain pattern, and that the resulting initial ideal is the same as in the classical case. In comparison with other previously studied variants of determinantal ideals, this appears to yield a rather distinct direction of generalization.
As mentioned previously, using the construction given in [Oss14a], we conclude from Theorem 1.3 that spaces of limit linear series are flat and Cohen-Macaulay; see Theorem 3.1 below. We then conclude in Corollary 3.3 that the two scheme structures on spaces of limit linear series agree under typical circumstances (specifically, when the space has the expected dimension, the open subset of refined limit linear series is dense, and the Eisenbud-Harris scheme structure is reduced). Finally, for the arguments of [CLT] and [CLPT], the crucial consequence is Corollary 3.4. This says that under the same conditions as Corollary 3.3, given a one-parameter smoothing of a curve of compact type, with generic fiber , we have a flat, proper moduli space whose special fiber is the Eisenbud-Harris moduli space of limit linear series on , and whose generic fiber is the usual linear series moduli space on .
2. Linked determinantal loci
We consider the following situation:
Let be a ring, and . Given , and nondecreasing integers , let be the polynomial ring over in variables with and .
For , let be the matrix over defined by
Given , let be the ideal of defined by all minors of all the . In addition, let be the universal determinantal ideal obtained from the minors of the matrix with entry equal to .
Consider the case , , , , , , , , and . Then the matrices are as follows:
We do not list every minor from each of the , but rather examine a selection of them with representative behavior. The minors from the first three columns of the are
respectively. The minors from the second, third and fourth columns of the are
respectively. Finally, the minors from the last three columns of the are
The key aspects of these minors are the following:
For each fixed minor position, the corresponding minors of the different are always equal to zero or a unique nonzero value.
The unique nonzero value always occurs for some , and always contains the “main diagonal” term of the usual universal minor.
There need not be a unique choice of achieving the unique nonzero minor value, and no single value of generates nonzero minors in all positions.
Note that the uniqueness of the nonzero minor value occurs in spite of the fact that a given nonzero minor may be generated by different patterns of zeroing out entries (as occurs with and for the minor from the second, third and fourth columns).
Throughout this section, we will use the lexicographic monomial order, induced by the ordering that comes before if or and . In order to prove Theorem 1.3, the main result is then the following:
In the case is a field, then the initial ideal of coincides with the initial ideal of , and in particular the Hilbert functions coincide. If , then also .
We prove this by exhibiting as a flat degeneration of , using the following construction.
Suppose , and let be the matrix over defined by
Thus, and .
In the context of Example 2.3, we have
If we consider the minor from the first three columns of , we get
Observe that if we factor out from this and then set we obtain the nonzero minor from of Example 2.3.
The minor from the second, third and fourth columns of is
and again factoring out and setting recovers the corresponding nonzero minors from , and .
Finally, the minor from the last three columns of is
and factoring out and setting yields the corresponding nonzero minor from .
The above, together with similar analysis of the remaining minors, says that in this example the ideal is governed in a suitable sense by the single matrix , and it follows that is a flat degeneration of the universal determinantal ideal .
In the case that is a field and , then if we fix with and with , let be obtained from the minor of by factoring out the largest possible power of , and then setting .
Then for all , the minor of is either equal to or to , and the latter occurs at least once. In addition, contains the monomial .
For each , let be the matrix obtained from by taking the entries with and , so that the minor we are considering is . Let be the number of such that , and the number of such that , so that . Then for any , there are so that the first rows of the matrix end in zeroes, while the last rows begin with zeroes (reading left to right). Explicitly, we have
where here we should use the convention that and . In particular, we have for . Now, clearly can only be nonzero if and , and in fact, since the individual terms appearing in the determinant are distinct monomials, there cannot be any cancellation, so the converse holds as well. Then , so if we choose maximal with , and if , then we have , so in this case . On the other hand, if , then , so again .
Thus, it remains to show that for any such that and , we have that , and that contains the monomial . Now, both and are obtained by omitting some monomials from the determinant of the matrix with entries given by , so we can prove the desired statements by explicitly identifying which monomials are included in each. First, for we observe that since in the powers of in the first rows are nondecreasing from left to right, and in the last rows are nonincreasing from left to right, it is clear that (within the rows and columns determined by ) if we take the top entries from the first columns (and consequently the remaining entries from the last columns) we will simultaneously minimize the power of coming from the top rows and the bottom rows. In particular, the “diagonal” term obtained from achieves the minimal possible power of in the relevant minor, so contains , as claimed. Moreover, we see that in order for a general term coming from for to have the minimal power of , we must have equal (as an unordered set with repetitions) to , and similarly for and .
Now, suppose that is not equal to . Then let be minimal such that the entry in column is taken from one of the bottom rows, so that occurs strictly fewer times in the sequence than in . If , we conclude from minimality of that for some , we have
On the other hand, if , we have that
and because we have taken the entry in the column from the bottom rows, we conclude that for some , we have
To summarize, we obtain a non-minimal power of for a given term if and only if for all , and for all . Thus, if we set so that , and so that , then we are saying simply that a given term appears in if and only if
Now we consider the monomials in for such that and . It is clear that a given monomial from entries occurs in if and only if
To compare (2.1) to (2.2), we note that is equivalent to saying that , and is the same as , which is the same as , so we can conclude that and , which is to say that the relevant range for is . It then follows immediately that (2.1) implies (2.2), and we wish to verify the converse.
If , the converse is likewise immediate. However, we see from the definitions that if , then , and then the first part of (2.1) is equivalent to the second part. In addition, we must have either or . In the first case, we have that , so the first part of (2.2) implies the first part of (2.1), which then implies the second part of (2.1) as well. But in the second case, the second part of (2.2) implies the second part of (2.1), which then implies the first part as well. ∎
Proof of Theorem 2.4.
First suppose that . We see that in this case, for the matrix may be obtained from by multiplying the righthand columns by , and dividing the bottom rows by . Thus, any given minor of is a power of times the corresponding minor of , and we concude that is simply equal to the ideal generated by the minors of . As this is obtained from by rescaling the variables by powers of , we find that , and also that the initial ideals and Hilbert functions coincide.
Now, suppose that . According to Lemma 2.7, in this case is generated by the , and the initial terms of the latter agree with the initial terms of the minors generating . Now, let be the flat degeneration of defined as in §15.8 of [Eis95] by the weight function assigning to the variable if and to the variable if . Then, it is clear from the definitions of the and of that for each . But by Theorem 15.17 and Exercise 20.14 of [Eis95], we have that the Hilbert functions of and of coincide, so we conclude that the Hilbert function of is less than or equal to the Hilbert function of . On the other hand, the universal minors form a Grobner basis for (see Theorem 5.3 of [BC03]), and their initial terms agree with those of the , so we conclude that . Again using invariance of Hilbert functions under flat degenerations, we conclude that the Hilbert function of is greater than or equal to that of , so they must be equal, and then we also have , as desired. ∎
Theorem 1.3 then follows by standard reductions to known results on the initial ideal of universal determinantal ideals. Indeed, we first conclude:
In the case is a field, then is reduced and Cohen-Macaulay, with codimension in . If further then is integral.
It is well known that is reduced and Cohen-Macaulay, of dimension ; see for instance Theorems 1.10, 5.3 and 6.7 of [BC03]. We conclude the same statements for by Theorem 2.4, together with Proposition 3.12 of [BC03]. In addition, for we have , and the latter is integral (again by Theorem 1.10 of [BC03]). ∎
We next find:
In the case that and , then is flat over , and integral and Cohen-Macaulay, with codimension in .
From Theorem 2.4 we have that all fibers of over have the same Hilbert function. Since is reduced, it follows from Exercise 20.14 of [Eis95] that is flat over . Given flatness, the statements on irreducibility and codimension follows from the corresponding statements on the generic fibers, which is a consequence of the statement of Corollary 2.8. Again using flatness, the statements on Cohen-Macaulyness and reducedness follow from the corresponding statements on fibers (see the Corollaries to Theorems 23.3 and 23.9 of [Mat86]), which is again Corollary 2.8. ∎
In order to deduce our main theorem from the universal case, we recall Lemma 2.3 of [OT14].
Suppose that is -linked. Let for , and by convention set , . Also, for set , and for set . Then locally on , for there exist subbundles of rank such that:
For we have that
and similarly .
For all , the restriction of to is an isomorphism onto a subbundle of , and for the restriction of to is an isomorphism onto a subbundle of .
The natural map
is an isomorphism for each .
Proof of Theorem 1.3.
The statement is local on , so we may assume that we have as in Lemma 2.10, and further that is affine and the are free. Choose bases of the and use them to induce bases of the via Lemma 2.10 (iii), but with reversed ordering (so that basis elements from come first, and those from last). Choosing arbitrary bases of and , we then have that the induced maps
are given by matrices of the form of our . These can be viewed as induced by a map , and our linked determinantal locus is then the pullback of the universal one under the corresponding morphism . The theorem then follows from Corollary 2.9 by Theorem 3 and Proposition 4 of [EN67] (see also the introduction of [HE71]). ∎
We conclude with a couple of examples showing that the definition of linked determinantal locus is somewhat delicate, in that minor variations will invalidate the conclusion of Theorem 2.4.
We first observe that a very similar pattern of zeroing out entries in a sequence of matrices can violate the uniqueness of nonzero minors proved in Lemma 2.7. Indeed, in Example 2.3, if we set to be instead of , the minor from the last three columns is , which does not agree with the corresponding minor from . Moreover, taking the difference yields , and since neither of these monomials is in the initial ideal of , we see that the conclusion of Theorem 2.4 is also violated in this case.
We also see that if we zero out monomials from the generators of the standard universal determinantal ideal, even if the initial terms of each generator remain unchanged, in general the initial ideals can change. For instance, if we let be the ideal generated by the minors of the matrix
and we let
then the initial terms of the generators are the same, and since we know the minors are a Grobner basis for , we conclude that
However, in this case the containment is strict: we see that
3. Applications to limit linear series
We now apply our results to draw conclusions on spaces of limit linear series. Because our results can be applied directly to limit linear series moduli space constructions carried out in [Oss14a], which are explicitly in terms of linked determinantal loci, we have elected to keep the presentation brief and not recall the rather lengthy definitions leading up to the aforementioned constructions. However, we will below briefly recall the Eisenbud-Harris limit linear series definition, so that all objects relevant to our final conclusion, Corollary 3.4, have been defined.
Our main theorem deals with moduli spaces of limit linear series on a one-parameter family of curves somewhat more general than those of compact type. In the general setting, there will be some extra data denoted by and , and an associated family , but in the compact type case, these are irrelevant, and in particular we may assume . In either case, we denote the closed point of by , and the special fiber of by . We will simultaneously treat the case of a single curve, where .
Suppose that we are in the situation of Theorem 6.1 of [Oss14a] (in particular, with the spectrum of a DVR), or of Theorem 5.9 of [Oss14a] (in which case is a point). Suppose also that the space of limit linear series on has the expected dimension at a given point . Then the limit linear series moduli space of Definition 6.3 of [Oss14a] is Cohen-Macaulay at , and flat over .
In fact, the theorem applies also to higher-dimensional base schemes – see Remark 3.5.
According to Proposition 6.4 of [Oss14a], is described by the construction in the proof of Theorem 6.1 of [Oss14a]. This construction proceeds by constructing as a closed subscheme of a scheme which is smooth over . Furthermore, is regular, so is likewise regular. The construction is given as an intersection of local equations ensuring vanishing along an auxiliary divisor , together with linked determinantal loci, each of expected codimension . The numbers work out that in order for to have dimension at , the aforementioned conditions must intersect with maximal codimension at . It thus follows that each individual condition cuts out a closed subscheme of of maximal codimension at , and therefore by Theorem 1.3 we conclude that each of these closed subschemes is Cohen-Macaulay at . By Lemma 4.4 of [HO08] we conclude that the intersection is likewise Cohen-Macaulay at .
Flatness is nontrivial only in the case that is positive-dimensional. Then Theorem 6.1 of [Oss14a] implies that is (universally) open over at , so every irreducible component containing dominates . Moreover, Cohen-Macaulayness implies that there are no imbedded components meeting , so since is the spectrum of a DVR, we conclude that is flat over at , as desired. ∎
We now recall the Eisenbud-Harris definition in the compact type case.
Given a curve of compact type, with dual graph , for let denote the corresponding component of , and for , let denote the corresponding node.
Given , a limit linear series (or more specifically, a limit ) on consists of a tuple of s on the components , satisfying the condition that for each connecting vertices , we have:
where denotes the vanishing sequence of at .
We say the limit linear series is refined if (3.1) is an equality for all .
In the compact type case, the Eisenbud-Harris definition of limit linear series leads to a natural scheme structure on the moduli space of limit linear seres, as a union of closed subschemes ranging over all possible refined ramification conditions at the nodes. This definition has the advantage of being very amenable to calculations, for instance in verifying reducedness. The alternative definition introduced in §4 of [Oss14b] and used in the statement of Theorem 3.1 also gives a scheme structure, which in principle could be different. The difficulty in comparing them arises from the union in the Eisenbud-Harris case, as the functor of points of a union cannot be easily described. However, we can now conclude that under typical circumstances, the two scheme structures agree.
If is a curve of compact type, suppose that we have the following conditions:
has the expected dimension ;
the refined limit linear series are dense in ;
the Eisenbud-Harris scheme structure on is reduced.
Then the Eisenbud-Harris scheme structure on coincides with the scheme structure introduced in [Oss14b].
According to Theorem 3.1, condition (I) implies that the scheme structure of [Oss14b] is Cohen-Macaulay. Thus, in view of condition (III), it is enough to show that the two scheme structures agree on a dense open subset, since Cohen-Macaulayness will then imply reducedness. But Proposition 4.2.6 of [Oss14b] asserts that the two scheme structures agree on the refined locus, so the desired result follows from condition (II). ∎
In the situation of Theorem 6.1 of [Oss14a], suppose further that the special fiber is of compact type and the conditions of Corollary 3.3 are satisfied. Then is flat and proper over , with special fiber equal to the Eisenbud-Harris scheme structure on , and generic fiber equal to the classical space of linear series on .
The moduli space construction of [Oss14a] restricted to the case that was the spectrum of a DVR partly to avoid developing technical hypotheses such as the “almost local” condition of §2.2 of [Oss14b], and partly because in the non-compact-type case, it is necessary to impose more stringent conditions on the family to ensure that every component of the curve comes from a divisor in the total family. With that said, under suitable hypotheses the construction does generalize to higher-dimensional base schemes , and in this case, Theorem 3.1 also generalizes. The only place where we used that was the spectrum of a DVR was in arguing flatness, but as long as is regular this argument can be replaced via the use of Theorem 14.2.1 of [GD66] and Proposition 6.1.5 of [GD65].
In the compact type context, it also suffices to assume that our family of curves has a single section (as opposed to a section through each component of the special fiber, as assumed in [Oss14a]). The desired result being etale local, we can then use an etale base change to produce sections through all components of the special fiber.
- [BC03] Winfried Bruns and Aldo Conca, Grobner bases and determinantal ideals, Commutative algebra, singularities and computer algebra (Siniai 2002), 2003, pp. 9–66.
- [CLPT] Melody Chan, Alberto López Martin, Nathan Pflueger, and Montserrat Teixidor i Bigas, Genus of the Brill-Noether curve, in preparation.
- [CLT] Abel Castorena, Alberto López Martin, and Montserrat Teixidor i Bigas, Invariants of the Brill-Noether curve, preprint.
- [EH86] David Eisenbud and Joe Harris, Limit linear series: Basic theory, Inventiones Mathematicae 85 (1986), no. 2, 337–371.
- [EH87a] by same author, Existence, decomposition, and limits of certain Weierstrass points, Inventiones Mathematicae 87 (1987), 495–515.
- [EH87b] by same author, The Kodaira dimension of the moduli space of curves of genus , Inventiones Mathematicae 90 (1987), 359–387.
- [Eis95] David Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, 1995.
- [EN67] John Eagon and Douglas Northcott, Generically acyclic complexes and generically perfect ideals, Proceedings of the Royal Society. Series A 299 (1967), 147–172.
- [GD65] Alexander Grothendieck and Jean Dieudonné, Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, seconde partie, Publications mathématiques de l’I.H.É.S., vol. 24, Institut des Hautes Études Scientifiques, 1965.
- [GD66] by same author, Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, troisième partie, Publications mathématiques de l’I.H.É.S., vol. 28, Institut des Hautes Études Scientifiques, 1966.
- [HE71] Melvin Hochster and John Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, American Journal of Mathematics 93 (1971), 1020–1058.
- [HO08] David Helm and Brian Osserman, Flatness of the linked Grassmannian, Proceedings of the AMS 136 (2008), no. 10, 3383–3390.
- [Kho] Deepak Khosla, Moduli spaces of curves with linear series and the slope conjecture, preprint.
- [Mat86] Hideyuki Matsumura, Commutative ring theory, Cambridge University Press, 1986.
- [Oss14a] Brian Osserman, Limit linear series for curves not of compact type, preprint, 2014.
- [Oss14b] by same author, Limit linear series moduli stacks in higher rank, preprint, 2014.
- [OT14] Brian Osserman and Montserrat Teixidor i Bigas, Linked alternating forms and linked symplectic Grassmannians, International Mathematics Research Notices 2014 (2014), no. 3, 720–744.