The Gieseker-Petri theorem and imposed ramification
We prove a smoothness result for spaces of linear series with prescribed ramification on twice-marked elliptic curves. In characteristic , we then apply the Eisenbud-Harris theory of limit linear series to deduce a new proof of the Gieseker-Petri theorem, along with a generalization to spaces of linear series with prescribed ramification at up to two points. Our main calculation involves the intersection of two Schubert cycles in a Grassmannian associated to almost-transverse flags.
The classical Brill-Noether theorem states that if we are given , a general curve of genus carries a linear series of projective dimension and degree if and only if the quantity
is nonnegative [GH80]. Moreover, in this case the moduli space of such linear series has pure dimension . This statement was generalized by Eisenbud and Harris to allow for imposed ramification: given marked points , and sequences for , consider the moduli space parametrizing linear series with vanishing sequence at least at each of the . Then Eisenbud and Harris used their theory of limit linear series to show in [EH86] that in characteristic , if is a general -marked curve of genus , the dimension of —if it is nonempty—is given by the generalized formula
The condition for nonemptiness is still combinatorial, but becomes more complicated in this context.
This theorem fails in positive characteristic for , but is still true if . In this case, we also have a simple criterion for nonemptiness. To state it, we shift notation, supposing we have marked points , and sequences , . We then introduce the following notation:
We summarize what was previously known about the space , as follows.
Given nonnegative integers, and sequences , , let be a twice-marked smooth projective curve of genus over a field of any characteristic. Set and set .
Suppose that and are general. Then is nonempty if and only if , and if nonempty, has pure dimension . Furthermore, it is reduced and Cohen-Macaulay, and if , it is connected.
For the nonemptiness and dimension statements, see [Oss14]; for reducedness and connectedness, see [Oss]. The Cohen-Macaulayness statement follows from the construction of (see for instance the proof of Proposition 3.1 below) together with the Cohen-Macaulayness of relative Schubert cycles.
What has remained open until now is the question of the singularities of the space . In the absence of marked points, Gieseker in 1982 used degenerations to prove a conjecture of Petri that if is general, then the space is also smooth [Gie82]. This proof was later simplified by Eisenbud and Harris [EH83] and Welters [Wel85] using ideas closely related to the theory of limit linear series. These proofs all relied on proving injectivity of the Petri map, by taking a hypothetical nonzero element of the kernel, and carrying out a careful analysis of how it would behave under degeneration.
In this paper, we give a new proof of the Gieseker-Petri theorem, and generalize it to the space , proving that the singular locus of this space consists precisely of linear series with a certain type of excess vanishing. Our Gieseker-Petri theorem with imposed ramification, Theorem 1.4 below, generalizes two statements.
In the absence of marked points, it reduces to the Gieseker Petri theorem, which holds for curves of any genus.
With marked points allowed, but in genus , it reduces to the well-known characterization of the singular loci of Schubert varieties and Richardson varieties.
Indeed, in the case , a single ramification condition corresponds to a Schubert cycle in the Grassmannian , while a pair of ramification conditions similarly corresponds to a Richardson variety. These spaces are singular, and their singularities can be characterized precisely as loci with a specific type of excess vanishing. Our main theorem extends this characterization to all genera, and also deduces additional consequences on the geometry of . To state it, the following preliminary notation will be helpful.
If is a on a smooth projective curve , and is an effective divisor on , write
Thus, to say that has vanishing sequence at least at is equivalent to saying that
We then make the following definition:
We see that contains all linear series with precisely the prescribed vanishing at and , but it also contains many linear series with more than the prescribed vanishing. For instance, if are both minimal, so that , then we also have . Our main theorem is then the following.
In the situation of Theorem 1.1, suppose further that we are in characteristic . Then the smooth locus of is precisely equal to .
Furthermore, the space has singularities in codimension at least , is normal, and when is irreducible.
Thus, we are in particular giving a new proof of the Gieseker-Petri theorem (in characteristic ). As an immediate consequence of Theorem 1.4, the twice-pointed Brill-Noether curves studied in [CLPT], as well as twice-pointed Brill-Noether surfaces [CP, ACT17] are smooth.
Our proof proceeds by degenerating to a chain of elliptic curves, and studying the geometry of the corresponding moduli space of Eisenbud-Harris limit linear series. The key idea in this step is that although the space of limit linear series will be singular in codimension , after base change and blowup one can ensure that any given point of on the generic fiber will specialize to a smooth point of the limit linear series space of a chain of curves of genus 0 or 1. This is where the characteristic- hypothesis comes in. The case of genus 0 is well-known, so our main calculation is the following result, which does not depend on characteristic, concerning the case .
In the situation of Theorem 1.1, suppose that , and make the generality condition explicit as follows: is arbitrary, and are such that is not a torsion point of of order less than or equal to . Then the space is smooth.
The proof of Theorem 1.5 proceeds by consideration of the morphism
The main subtlety that needs to be addressed is that the map (1.2) is not smooth. The fibers are each described as an intersection of a pair of Schubert cycles in a Grassmannian. But in finitely many fibers, namely the ones above line bundles of the form for , the pairs of flags defining the Schubert cycles are not transverse, but only almost-transverse (see Definition 2.7). We prove Theorem 1.5 by first showing that in fibers, the tangent spaces have dimension at most greater than expected, and then showing that at the points where the tangent space dimension jumps in the fiber, there cannot be any horizontal tangent vectors.
The statement on tangent spaces in fibers, which is Corollary 2.12 below, takes place entirely inside the Grassmannian, and may be of independent interest. Indeed, §2 is a study of tangent spaces of intersections of pairs of Schubert cycles, and we address the case of arbitrary pairs of flags in Theorem 2.10 and Remark 2.13. We then prove Theorems 1.4 and 1.5 in §3.
2. Almost-transverse intersections of Schubert cycles
It is well known that the intersection of two Schubert varieties associated to transverse flags—commonly called a Richardson variety—is smooth on the open subset of points which are smooth in both Schubert varieties. In this section we consider intersections of pairs of Schubert varieties associated to not necessarily transverse flags. Our analysis recovers the usual smoothness statement in the transverse case, but our main purpose is to analyze the almost-transverse case in Corollary 2.12, where we characterize the smooth points and show that the dimension of the tangent space jumps only by at the non-smooth points. While Schubert intersections and non-transverse flags have been studied by Vakil [Vak06] and Coskun [Cos09], those situations involved studying the flat limits of transverse intersections, rather than the direct analysis of the non-transverse intersections required in the present work.111More precisely, they work with closures of loci with prescribed behavior with respect to both flags; these are in particular irreducible, so are not the same thing as the intersection of Schubert cycles which we consider.
We fix to be an algebraically closed field of any characteristic. Throughout this section, we will work entirely with -valued (equivalently, closed) points. We index our complete flags by codimension, so that for a complete flag in a -dimensional vector space ,
We fix further notation as follows.
Given a -vector space of finite dimension and a complete flag in , if we are given also with
we let be the Schubert variety defined as the closed subscheme of given by the set of such that
More precisely, the conditions in (2.1) are determinantal, yielding a scheme structure on (which turns out to be reduced). In our notation, the codimension of is given by .
With an increasing sequence as above, say that an index with is active in if and , or and
Let be the open subscheme of consisting of subspaces for which for every active index , the inequality in (2.1) is an equality.
We fix the following situation throughout this section.
Let be a finite-dimensional -vector space, and write . Fix complete flags , in , and sequences with and
Recall that we are indexing by codimension; thus . Note that for any , the distinct subspaces in the collection form a complete flag in ; we denote the flag by abuse of notation.
We have the following description of the tangent space at any point in . Tangent spaces to Schubert varieties are well understood [BL00], but for the sake of completeness, we provide a description in the particular case that we need of Grassmannian Schubert varieties.
Given , let be the set of active indices such that . Then there is a canonical isomorphism of vector spaces
In particular, the smooth locus of is precisely .
By definition, is the scheme-theoretic intersection of the following subschemes of (for ):
Define to be the open subscheme of where equality holds. Note also that in fact can be cut out as the intersection of the over all active indices : this is immediate set-theoretically, and is also true scheme-theoretically because whenever , the condition for is obtained from that of by adding a single row to the local matrix expression, and considering minors of size one larger. Thus, every minor occuring in the condition can be expanded in terms of minors occuring in the condition.
The first statement of the proposition then follows immediately from the following claim. For a fixed index ,
where we identify with as usual.
To prove this claim, one may work on an affine open subset of , as follows. Choose a basis of extending a basis of ; then an affine neighborhood of is given by the set of matrices whose first columns form the identity matrix (where the point in is given by taking the span of the rows). More precisely, for any -algebra , we may identify the -points of this open subscheme with -valued matrices whose first columns form the identity matrix. In particular, taking , the tangent space is identified with matrices in block form , where is a matrix of values of ; the matrix then determines an element of . Now, we may further assume that the chosen basis of also includes a basis of as a subset. Then the -points of consist of those matrices such that the submatrix consisting of all columns not corresponding to the basis of has rank at most . Assuming that we order our basis of so that a basis of comes at the end, the submatrix in question has the form
where the size of the identity matrix in the upper left is . Therefore, lying in corresponds to the condition that
Now, specialize to the case , and consider a tangent vector to at . The submatrix is a multiple of . Therefore all and larger minors of are guaranteed to vanish. Thus in the case (i.e. ), all tangent vectors to at are also tangent vectors to at . On the other hand, when , a tangent vector to at is a tangent vector to if and only if the matrix vanishes entirely. This condition can be made intrinsic by observing that, if is the linear map encoding a tangent vector, then is a matrix representation for the linear map induced by . Therefore it follows that, in the case , described a tangent vector to if and only if . This proves the claim, and the first statement of the proposition.
The second statement follows by direct computation of the codimension imposed by the conditions on the tangent space in the first part. If we have , let denote the next (greater) element of , setting if is maximal in . By starting from the condition imposed at the maximal element of , and inductively working downwards, one computes that the codimension of the tangent space is given by
Each term of this sum is always less than or equal to , with equality if and only if there are no actives indices strictly between and . The proposition follows. ∎
Given , there is a canonical isomorphism of vector spaces
Following Definition 4.1 of [CP], we define:
Two complete flags and are called almost-transverse if there exists an index such that
More generally, we have the following statement, which is easy to check:
There is a unique permutation associated to the flags and with the property that there exists a basis for satisfying
Such a basis can also be characterized by the property that for all indices and , is spanned by . In particular, if then contains one of the .
We refer to a basis as in Proposition 2.8 as a -basis.
Thus, following the notation of Proposition 2.8, we have that if and only if , and are transverse if and only if , and and are almost-transverse if and only if is the composition of with an adjacent transposition.
We set , so that
is precisely the expected dimension of . Also recall the definition of the complete flags and from Situation 2.4.
Given , let denote the permutation associated to and in by Proposition 2.8. Given any , let
setting if no such exists. Similarly, let
or if no such exists. Then
Let be a -basis for . Then for any active in , respectively , have
In other words, given any , and any that is active in , we have if and only if . Similarly, for any that is active in , we have if and only if . By Corollary 2.6, we have isomorphisms
We are thus reduced to computing the dimensions , which are equal to
Moreover, the first two terms are determined by the fact that and , by assumption that and are active or are equal to . A straightforward calculation produces (2.2). ∎
We observe that the well-known case of transverse flags follows immediately from Theorem 2.10.
If and are transverse, then for all .
Following the notation of the proof of Theorem 2.10, for each we have by construction. Since and are transverse, it follows that , so . ∎
More importantly, we can also deduce the desired statement in the almost-transverse case.
Given , suppose and are almost-transverse, with such that .
Suppose first that for active in , that for active in , and that
If those conditions do not all hold, then
Let be a -basis for . First suppose that for active in , and for active in , and that We will deduce that
Now, for , the fact that and and are almost-transverse implies that either , or that and . But the latter case cannot be, since then both , contradicting that . Therefore and
Next, to show that , we claim that and . Recall that and . By assumption, is active in and , so by (2.3). We want to show that is the largest active index in with . Indeed, if is active in with , then . But now , so . Therefore , contradiction. A similar argument shows . Therefore,
since is a hyperplane in , and is contained in it by assumption.
It remains to show that if the conditions in the statement of Corollary 2.12 do not all hold, then We prove the contrapositive. Suppose that By Theorem 2.10, there is an index such that . Again, given that and that and are almost-transverse, it follows that either or that and . But would imply , contradicting the codimension statement. So and , implying that and for active indices and in and respectively. (It is not possible that or , since .) Furthermore,
where the last containment holds again by the codimension assumption.
Summarizing, we have shown that the only way that is for all the conditions in the statement of Corollary 2.12 to hold, in which case we have already proved that the dimension is exactly . ∎
For arbitrary flags and and , let be the associated permutation from Proposition 2.8 (maintaining other notation as in Theorem 2.10). Then the extent to which the dimension of the tangent space at of exceeds can be bounded in terms of as follows. We have:
Here denotes the decreasing permutation , and denotes the inversion number of , i.e. the number of with .222If one views as a Coxeter group with reflections being the adjacent transpositions, then is also the Coxeter length of . We briefly sketch a proof of this more general inequality.
Using the second part of Proposition 2.8, it follows that for each , we have
From this it follows that if and only if . Now, since we have , we find that . Note also that for all , and . Then:
3. Linear series in positive genera
We begin with a proposition that will show that our smoothness result, Theorem 1.4 to be proved below, is sharp.
In the situation of Theorem 1.1, every point of in the complement of is singular.
This is a consequence of the standard construction of the space : we let be a Poincaré line bundle on . Take a sufficiently ample333Precisely, of degree strictly greater than effective divisor on with support disjoint from and , and write . Writing for projection, let be the relative Grassmannian , equipped with structure map . Let denote the universal subbundle. Then is cut out in by the condition that the induced map
vanishes identically. Because we have chosen to have support disjoint from and to be sufficiently ample, the space is cut out by imposing the additional Schubert condition that the maps
have rank at most for each . Imposing the analogous condition at , we obtain as an intersection of three conditions: a determinantal condition (in fact a complete intersection), and two relative Schubert cycles. It is routine to check that for to have dimension , as asserted by Theorem 1.1, these three conditions must intersect in the maximal codimension. Given that we know from Theorem 1.1 that does in fact have dimension , it then further follows that in order for to be smooth at any point, that point must lie in the smooth locus of each of the three conditions, and in particular of the two Schubert cycles.
But we claim that consists precisely of the points of which lie in the smooth locus of both relative Schubert cycles. Indeed, each relative Schubert cycle is nothing but a locally constant family of Schubert varieties over the base , so we are done by the standard characterization of the smooth locus of a Schubert variety (see the second part of Proposition 2.5). ∎
We now use our calculations in Grassmannians in §2 to complete the proof of our main theorem, beginning with the case of genus in Theorem 1.5.
Proof of Theorem 1.5.
Set . We may assume , as otherwise the result is trivial. Thus, is a Grassmannian bundle over , with the fiber over a line bundle being canonically identified with . The condition imposed by requiring vanishing sequence at least at then gives a Schubert cycle in each fiber, corresponding to the complete flag determined by vanishing order at . The codimension of spaces in the flag corresponds precisely to vanishing order except over the point , where no sections vanish to order precisely , and vanishing to order imposes codimension only . Consequently, there are two possibilities for . First, if , it is a relative Schubert cycle of codimension in , Cohen-Macaulay and flat over . Or, if , it is supported entirely over (even scheme-theoretically), and is still a Schubert cycle, but of codimension . The same analysis applies to , so we find that every fiber of
is an intersection of a pair of Schubert cycles. The basic properties of the map (3.1) are analyzed for instance in Lemma 2.1 of [Oss14] and Proposition 2.1 of [Oss]; we review the main points of this analysis in order to carry out the necessary tangent space analysis.
First, we see that in most fibers of (3.1), the relevant Schubert cycles are associated to transverse flags: the only way in which the flags fail to be transverse is if for some , which is unique by genericity of and ; then the conditions of vanishing to order at and at intersect in dimension instead of dimension . Thus, on fibers of (3.1) over points not of the form for , we have that the Schubert indexing matches vanishing sequences, and the flags are transverse, so the standard theory (see for instance Corollary 2.11) gives us that (on these fibers) the space is smooth (of relative dimension ) over , and hence smooth of relative dimension over . Similarly, it is easily verified that we still obtain Richardson varieties over