Schreyer's Conjecture on Canonical Curves

• Schreyer's Conjecture is a refinement of Green's Conjecture, predicting that the top linear syzygies of a canonical curve are determined by the rational normal scroll formed by its unique minimal pencil.
• It asserts that for a general k-gonal curve, the final nonzero syzygy has dimension g–k, arising from the Eagon–Northcott complex linked to the scroll geometry.
• The conjecture connects the algebraic structure of the minimal free resolution with geometric features, thus encoding gonality and special linear series in the curve's canonical Betti table.

Searching arXiv for relevant papers on Schreyer's Conjecture and related syzygy results. Search query: "Schreyer's Conjecture canonical curves gonality" Schreyer’s Conjecture is a refinement of Green’s Conjecture for canonically embedded curves. It predicts that, for a smooth curve $C$ of genus $g$ and non-maximal gonality $k$, the highest-order linear syzygies of the canonical embedding are completely determined by the $(k-1)$-dimensional rational normal scroll swept out by the unique minimal pencil $g^1_k$, and that the top linear Betti number equals $g-k$ (Farkas et al., 2016). In the formulation emphasized by later work, the conjecture belongs to a broader syzygy program initiated by Schreyer, in which the minimal free resolution of the canonical ring records the existence, uniqueness, and geometry of special linear series through scrollar syzygies and their Eagon–Northcott origin (Bopp et al., 2018).

1. Canonical curves, Betti numbers, and the linear strand

Let $C$ be a smooth nonhyperelliptic curve of genus $g$. The complete linear series $|K_C|$ defines the canonical embedding

$C \to \mathbb{P}^{g-1}.$

Writing

$g$0

and $g$1 for the coordinate ring of the canonically embedded curve, one studies the minimal graded free resolution of $g$2. Since $g$3 is Gorenstein of codimension $g$4, this resolution is self-dual and has length $g$5, with graded Betti numbers

$g$6

or, equivalently for canonical curves,

$g$7

via Koszul cohomology (Bopp et al., 2018).

In the notation of (Farkas et al., 2016), the linear strand is the row $g$8, with entries

$g$9

Green’s Conjecture describes the vanishing pattern of these groups in terms of the Clifford index. Schreyer’s Conjecture addresses the final nonvanishing position and asserts more than mere vanishing: it identifies the dimension and geometric source of the top linear syzygies (Farkas et al., 2016).

The key geometric input is gonality. If $k$0 carries a base-point-free pencil $k$1, equivalently a degree-$k$2 morphism $k$3, then the fibers of $k$4 sweep out a rational normal scroll $k$5 of dimension $k$6. The Eagon–Northcott complex resolving the scroll produces canonical linear syzygies, and Schreyer’s Conjecture asserts that, in the relevant top degree, these scroll syzygies account for all linear syzygies of the curve (Farkas et al., 2016).

2. Statement of the conjecture and its syzygetic meaning

In the form proved for general curves of prescribed gonality, Schreyer’s Conjecture concerns a smooth curve $k$7 of genus $k$8 with non-maximal gonality $k$9. Under the hypotheses that $(k-1)$0 is a reduced single point and that $(k-1)$1 is the unique line bundle of degree $(k-1)$2 computing the Clifford index, the conjecture predicts

$(k-1)$3

and that all highest-order linear syzygies are of Eagon–Northcott type, induced by the rational normal scroll determined by the unique minimal pencil $(k-1)$4 (Farkas et al., 2016).

This formulation strengthens Green’s Conjecture. Green’s Conjecture determines the last nonvanishing index of the linear strand: for a general $(k-1)$5-gonal curve one has $(k-1)$6 if and only if $(k-1)$7, so the linear strand stops at $(k-1)$8. Schreyer’s Conjecture then specifies the size and structure of the final nonzero piece, not only asserting nonvanishing at $(k-1)$9 but identifying its exact dimension $g^1_k$0 and its origin in the scroll (Farkas et al., 2016).

The same phenomenon can be expressed in the second linear strand of the self-dual resolution. In the notation of (Bopp et al., 2018), the Schicho–Schreyer–Weimann criterion states that for a smooth canonically embedded curve $g^1_k$1 of genus $g^1_k$2 and $g^1_k$3,

$g^1_k$4

and

$g^1_k$5

This is the second-strand counterpart of the statement $g^1_k$6, obtained through Gorenstein duality. It encodes the same principle: the unique minimal pencil leaves a precise “scrollar” footprint in the canonical resolution (Bopp et al., 2018).

3. Scrolls, minimal pencils, and Eagon–Northcott syzygies

The geometric mechanism behind Schreyer’s Conjecture is the containment of the canonical curve in a rational normal scroll. If $g^1_k$7 is a base-point-free $g^1_k$8, then $g^1_k$9 lies on a $g-k$0-dimensional rational normal scroll of degree $g-k$1, swept out by the spans of fibers of the map $g-k$2 defined by $g-k$3 (Farkas et al., 2016). In the terminology emphasized by (Bopp et al., 2018), this scroll is the geometric support of the expected gonality syzygies.

The Eagon–Northcott complex resolving the scroll contributes a distinguished block of linear syzygies. Schreyer’s program predicts that, for a general $g-k$4-gonal curve with unique minimal pencil, these are exactly the highest-order linear syzygies of the curve. In particular, the number of such “scrollar” syzygies is $g-k$5, which yields the predicted equality $g-k$6 or, equivalently in the second strand, $g-k$7 (Farkas et al., 2016, Bopp et al., 2018).

A more explicit description appears in the sufficient criterion of (Farkas et al., 2016). Assuming the bpf-linear growth condition, uniqueness of the minimal pencil with simple ramification, and $g-k$8, there is a canonical identification

$g-k$9

and all syzygies in $C$0 are of Eagon–Northcott type (Farkas et al., 2016). This identifies the vector space of top linear syzygies with a tensor construction built directly from the minimal pencil.

The conceptual significance is that the canonical resolution becomes a detector for special linear series. A unique minimal pencil gives rise to a unique scroll, the scroll contributes an Eagon–Northcott complex, and the top linear syzygies are predicted to be precisely those inherited from that complex. This is the sense in which Schreyer’s Conjecture goes beyond a vanishing theorem: it is a structural statement about the geometric origin of syzygies (Farkas et al., 2016).

4. Proof for general curves of non-maximal gonality

The principal theorem of (Farkas et al., 2016) establishes Schreyer’s Conjecture for general curves of prescribed gonality. Specifically, if $C$1 is a general $C$2-gonal curve of genus $C$3, equivalently $C$4, then

$C$5

and the highest-order linear syzygies are of Eagon–Northcott type, induced by the rational normal scroll determined by the unique minimal pencil $C$6 (Farkas et al., 2016).

The proof has several layers. The divisorial base case $C$7 uses the relation between the Koszul divisor and the Hurwitz divisor on $C$8, yielding the equality $C$9 when the minimal pencil is unique and $g$0 (Farkas et al., 2016). In higher genus, the argument proceeds by degenerating a general $g$1-gonal curve to a nodal curve in the divisorial range, then analyzing the corresponding admissible cover in the Hurwitz space.

A central role is played by the Eagon–Northcott divisor $g$2, which parametrizes covers whose canonical curves have extra top linear syzygies beyond the Eagon–Northcott ones, namely points with $g$3 (Farkas et al., 2016). The paper realizes $g$4 as the degeneracy locus of a morphism of vector bundles of equal rank over a moduli space of stable maps to $g$5, and then shows that the admissible covers arising from the degeneration of a general $g$6-gonal curve avoid this divisor. Boundary analysis is completed using K3 surfaces and syzygy arguments of Voisin type, showing by semicontinuity that the general curve cannot have extra top linear syzygies (Farkas et al., 2016).

The result verifies that, in the general non-maximal gonality range, the top of the linear strand is entirely controlled by the ambient scroll. This settles the conjectural picture for general $g$7-gonal curves in characteristic zero and supplies a precise geometric interpretation of the final nonvanishing linear syzygies (Farkas et al., 2016).

5. Relation to Green’s Conjecture and the Schicho–Schreyer–Weimann criterion

Schreyer’s Conjecture sits naturally between Green’s Conjecture and more refined scroll-detection statements. Green’s Conjecture for a smooth nonhyperelliptic curve $g$8 asserts that

$g$9

and

$|K_C|$0

Via Gorenstein duality, this can be rephrased as a vanishing pattern for the second linear strand, namely the vanishing of $|K_C|$1 up to the corresponding dual index (Bopp et al., 2018).

For general $|K_C|$2-gonal curves in characteristic zero, Aprodu proved Green’s Conjecture, and (Farkas et al., 2016) shows that Schreyer’s Conjecture refines the final nonvanishing position by identifying the exact dimension and Eagon–Northcott origin of the top linear syzygies. Thus Green’s Conjecture says where the linear strand ends, while Schreyer’s Conjecture says what occupies that final position (Farkas et al., 2016).

The Schicho–Schreyer–Weimann conjecture, as formulated in (Bopp et al., 2018), makes this refinement particularly concrete. It characterizes the case where the minimal pencil is unique and reduced by the two conditions

$|K_C|$3

In characteristic $|K_C|$4, this criterion was proven by Farkas and Kemeny, and (Bopp et al., 2018) explicitly interprets it as the syzygetic footprint of a unique minimal pencil: the curve lies on a rational normal scroll $|K_C|$5 of dimension $|K_C|$6 and degree $|K_C|$7, and the Eagon–Northcott complex of $|K_C|$8 contributes a block of linear syzygies of length $|K_C|$9 (Bopp et al., 2018).

This makes clear why Schreyer’s Conjecture is often described as going beyond Green’s Conjecture. It does not merely relate syzygies to the Clifford index. It asserts that the syzygies at the scrollar place determine, and are determined by, the minimal pencil and the associated rational normal scroll (Bopp et al., 2018).

6. Positive characteristic and the refined formulation

In positive characteristic, the characteristic-zero vanishing picture can fail because of extra syzygies. The computational study of (Bopp et al., 2018) documents this systematically and proposes a refined version of Green’s Conjecture designed to remain meaningful in small characteristic. Instead of demanding vanishing along the second linear strand, it requires finite-length homology in that strand up to the Clifford threshold.

The refined statement is as follows. Let $C \to \mathbb{P}^{g-1}.$0 be a canonically embedded curve, and let $C \to \mathbb{P}^{g-1}.$1 be the second linear strand of a minimal free resolution of $C \to \mathbb{P}^{g-1}.$2. Then:

• $C \to \mathbb{P}^{g-1}.$3 is a module of finite length for all $C \to \mathbb{P}^{g-1}.$4 if and only if $C \to \mathbb{P}^{g-1}.$5.
• If $C \to \mathbb{P}^{g-1}.$6 is general inside the gonality stratum $C \to \mathbb{P}^{g-1}.$7 with $C \to \mathbb{P}^{g-1}.$8, then $C \to \mathbb{P}^{g-1}.$9 is supported on the rational normal scroll swept out by the unique $g$00 on $g$01 (Bopp et al., 2018).

This replacement preserves the geometric core of Schreyer’s syzygy program. Even when extra syzygies appear and pure vanishing fails, the defect is predicted to be finite-length, while the genuine gonality contribution remains localized on the scroll swept out by the minimal pencil (Bopp et al., 2018). A plausible implication is that the refined formulation separates “spurious” characteristic-$g$02 syzygies from the scrollar syzygies that encode the curve’s linear series.

The paper records classical failures for general curves in small characteristic: genus $g$03 in characteristic $g$04 and genus $g$05 in characteristic $g$06 exhibit extra syzygies at the critical Betti number, contradicting the characteristic-zero naturality pattern (Bopp et al., 2018). It also reports additional exceptional pairs for $g$07, based on random canonically embedded curves constructed over finite fields. In these cases, however, the first nonzero Betti number in the second strand still occurs at the critical index, and the homology behaves as predicted by the refined conjecture (Bopp et al., 2018).

The genus $g$08, characteristic $g$09 experiments are especially illustrative. For $g$10 random genus $g$11 curves over $g$12, the first nonzero $g$13 is $g$14, with observed values $g$15, $g$16, and $g$17, among others. The corresponding annihilator data include pairs such as $g$18, $g$19, $g$20, and $g$21, where dimension $g$22 indicates support on a $g$23-dimensional scroll and degree a multiple of $g$24, suggesting multiple $g$25’s; the case $g$26 indicates finite-length homology (Bopp et al., 2018). These computations support the idea that small-characteristic anomalies need not destroy the underlying scroll detection predicted by Schreyer’s program.

7. Scope, consequences, and limitations

For general curves of non-maximal gonality in characteristic $g$27, Schreyer’s Conjecture is established: the top linear Betti number equals $g$28, and the corresponding syzygies are exactly those coming from the Eagon–Northcott complex of the rational normal scroll determined by the unique minimal pencil (Farkas et al., 2016). This yields a precise geometric interpretation of the last nonzero row of the canonical Betti table and confirms that the canonical resolution detects gonality in a stronger sense than Green’s Conjecture alone.

The conjecture is not intended as a universal statement for all curves with fixed gonality. The top Betti number can jump for special curves with multiple minimal pencils or with special Brill–Noether behavior. In the formulation cited by (Farkas et al., 2016), if $g$29 is not a reduced point, then $g$30. Likewise, (Bopp et al., 2018) emphasizes that in small characteristic extra syzygies can appear even for general curves, and these may not be explained by an actual linear series on the curve. This is precisely why the refined positive-characteristic formulation replaces strict vanishing by a finite-length homology condition.

At the same time, the theory supplies robust sufficient criteria beyond the generic case. Theorem 0.5 of (Farkas et al., 2016) proves Schreyer’s statement under the bpf-linear growth condition together with uniqueness, simple ramification, and the condition $g$31. In this range one obtains not only the equality $g$32 but also an explicit description of the top syzygies as tensors of sections of $g$33 and $g$34 (Farkas et al., 2016).

The broader significance of Schreyer’s Conjecture lies in the unification of syzygy theory and Brill–Noether geometry. Canonical Betti tables do not merely reflect abstract homological complexity; they encode the presence and uniqueness of minimal pencils, the associated scroll geometry, and, in refined positive-characteristic form, the distinction between scroll-supported syzygies and finite-length anomalies (Bopp et al., 2018). In this sense, Schreyer’s Conjecture is a central statement in the program of reading the geometry of a curve directly from the linear strands of its canonical resolution.