Brill-Noether theory of curves on toric surfaces
A Laurent polynomial in two variables naturally describes a projective curve on a toric surface. We show that if is a smooth curve of genus at least 7, then is not Brill-Noether general. To accomplish this, we classify all Newton polygons that admit such curves whose divisors all have nonnegative Brill-Noether number.
1. Introduction and statement of results
Let be a smooth projective curve of genus defined over a field . Brill-Noether theory is the study of certain special divisors on with unusually large complete linear systems, namely those divisors for which Riemann’s inequality is strict. Let be a divisor on of degree and rank . The Brill-Noether number associated to is then defined by the equation . The Brill-Noether Theorem of Griffiths and Harris indicates that a general family of curves, the so-called Brill-Noether general curves, have no divisors with negative Brill-Noether number.
In this paper we examine a particular family of curves, namely the smooth projective curves defined by some Laurent polynomial in two variables on the toric surface associated to the Newton polygon of . We determine that no such curve is Brill-Noether general if its genus is greater than six. Let be a Laurent polynomial in two variables , let be the Newton polygon associated to , the toric surface associated to , and the zero-set of in . Then we will establish the following result:
Let be the closure of the image of under the natural map . If is smooth and has genus at least 7, then is not Brill-Noether general. Moreover, if , then there exists a divisor on of rank 1 with negative Brill-Noether number.
This bounds of this theorem are sharp; we will exhibit examples of curves without these special divisors for each genus with .
In fact, we prove a stronger result, and classify the interior Newton polygons of such general curves up to transformations of the lattice with the following, more technical theorem.
There is an and associated to a Newton polygon with no divisors having negative Brill-Noether number if and only if the convex hull of the interior lattice points of is empty or transforms under some invertible lattice transformation to one of the 11 polygons in Figure 1. Furthermore, if the interior Newton polygon is not equivalent under a lattice transformation to any of these 11 and is also not equivalent to the polygons in Figure 2 or Figure 3, then the gonality of is less than .
This paper is organized as follows. In Section 2 we introduce the concepts and terminology from algebraic geometry and discrete geometry that will underlie the results in this paper. In Section 3 we use a bound on the lattice width of a curve’s Newton polygon to establish the non-existence of curves without special divisors in the case and limit the set of possible interior Newton polygons to those of Figure 1. We also handle the exceptional genus 10 case in Proposition 3.6. In Section 4 we will then establish the existence of curves having no divisors with in the eleven cases indicated by Theorem 1.2.
Let be a smooth projective curve of genus defined over a field . We refer to the dimension of the complete linear system of a divisor as the rank of . The Brill-Noether number of , given by , then gives an expected dimension of the subscheme of the Picard group of consisting of all degree divisor classes of rank at least .
The foundational result of Brill-Noether theory, the Brill-Noether Theorem of Griffiths and Harris, indicates that this expected dimension is correct for a general curve. We present a simplified version.
Theorem 2.1 ().
Let be a smooth projective curve of genus . Then the following statements are true:
If , then there exists a divisor on of degree and rank .
If is general and , then no divisor of degree on has rank .
From this theorem follows immediately that on a general curve no divisor has negative Brill-Noether number. Despite this, many curves that are easy to describe, such as complete intersections in projective space, tend to have divisors with negative Brill-Noether number, and constructing particular curves of high genus having no such divisors requires surprising effort. Recently, there has been a proof using tropical methods by Cools et al. in .
2.1. Lattice polygons and toric surfaces
A lattice polygon is defined to be a polygon in whose vertices have integer coordinates. If is a convex lattice polygon, then, following , we define the interior polygon of as the convex hull of the set of lattice points contained in the interior of .
A lattice transformation is an invertible linear map . They can be thought of as those transformations in . This notion allows us to define the lattice width of a polygon robustly, as follows.
The lattice width of a lattice polygon is the minimal width of the image of under a nontrivial linear map .
Lattice polygons in general are easy to analyze geometrically. One important result on general lattice polygons, both in general and in proving our main results, is the classical Pick’s theorem:
If is a lattice polygon, the area of is given by , where is the number of lattice points on the boundary of the polygon and is the number of points in the interior of .
More recent results on lattice polygons include the following bound on the area of a convex lattice polygon in terms of its lattice width.
Theorem 2.4 (, Theorem 2).
If is a convex lattice polygon, then the area of is at least .
Also, the lattice width of a lattice polygon’s interior polygon determines its own lattice width.
Theorem 2.5 (, Theorem 4).
The lattice width of is given by
unless is equivalent under lattice transformation to the convex hull of , and for some , in which case .
It is possible to associate to any convex lattice polygon a projective surface using the following construction, described in :
If is a convex lattice polygon containing lattice points, the toric surface associated to , denoted , is the Zariski closure in of the map given by .
Let be the lattice triangle with vertices at , , and , which thus also contains the interior point . The toric surface is then the closure of the image of under the map , which is the surface in defined by .
Let be a nonzero Laurent polynomial, and let be the curve of its zeros in . The Newton polygon of is defined as the convex hull in of the points with .
An important use of the Newton polygon of a curve is in calculating the genus of a smooth curve. Indeed, the following result holds.
Theorem 2.8 (, Section 4, Assertion 2).
Let be a Laurent polynomial with Newton polygon . Let be the closure of in . If is smooth, then the genus of is the number of interior points of .
Proof of Theorem 1.1.
Let be a Laurent polynomial in two variables, let be the Newton polygon of , and let be the closure of in . Suppose is smooth, and let be the genus of By Theorem 2.8, we have that the interior polygon of the Newton polygon associated to contains points. But by Theorem 1.2 we also have that is equivalent under some lattice transformation to one of the polygons in Figure 1, and every such lattice polygon contains at most six points, so we have . This proves the first half of the theorem.
Now suppose is a curve having no rank 1 divisors with negative Brill-Noether number. As before, by Theorem 2.8 we have that the interior polygon of the Newton polygon associated to to has points, and by Theorem 1.2 is equivalent to one of the lattice polygons in Figure 1, Figure 2, or Figure 3. Since those polygons all have either at most six or ten points, we hence have or , as was to be shown. ∎
3. Constructing special divisors
In this section we prove one half of Theorem 1.2; we show that if a Laurent polynomial in two variables with Newton polygon describes a smooth curve having no divisors with negative Brill-Noether number, then is equivalent under a lattice transformation to a polygon in Figure 1. To do so, we will bound the gonality of these curves.
The gonality of a curve is the minimal degree of a nonconstant rational map .
Because the zero-set of such a rational map is a divisor of rank 1, we can equivalently think of the gonality of as the minimal degree of a rank 1 divisor on .
Using the structure of the Newton polygon we can bound the gonality of the curves we are investigating. For the rest of the section, let be a Laurent polynomial in two variables with Newton polygon , the zero set of in , and the closure of the image of under the embedding of the torus in the toric surface associated to .
The gonality of is at most the lattice width of .
Suppose has lattice width that without loss of generality is realized by the linear map that sends to . Then the map given by has degree , for is generated over by , and can hence be expressed in the form for some integer and some Laurent polynomials in . As such, dividing out factors of gives us a degree polynomial for over , so the map has degree and has gonality at most . As such, since is birationally equivalent to , we have that has gonality at most , as was to be shown. ∎
We now use this fact and the properties of lattice polygons given in the previous section to prove a useful bound on the gonality.
Suppose is smooth and has genus . Then the gonality of is at most .
Suppose first that is not equivalent under lattice transformation to the convex hull of , and for any . Let be the gonality of . We have by Lemma 3.2 that is at most the lattice width of and hence (noting that we have excluded the exceptional case) Theorem 2.5 implies that . Now, there are precisely lattice points within or on the edges of . We note that at most of those points can be interior for any , for if is two-dimensional, it has at least three vertices, and if it is one-dimensional, it has zero interior vertices. As such, assuming , the area of is at most by Pick’s theorem. Therefore, using the lower bound for area provided by Theorem 2.4 we see that
As such, since , we have:
From this we conclude , as was to be shown.
Now, if is equivalent to the convex hull of , , and , then is the projective plane and is a smooth plane curve of degree . So has gonality and genus . Since , we hence have , and by direct calculation we have that , proving the result for all . ∎
If is smooth and has genus , then has a divisor of rank 1 and with negative Brill-Noether number.
Let be a smooth curve on which no rank 1 divisor has negative Brill-Noether number. As such, it has no divisors of rank 1 and degree , and hence has gonality at least . But then, from Theorem 3.3 we can conclude . This inequality is false when , establishing the result. ∎
With this result much of the work of the classification is done; we know we can restrict our attention to those lattice polygons containing twelve or fewer interior points. When the lattice polygon also does not contain ten interior points, this is particularly simple.
The lattice width of a convex lattice polygon with interior points, where and , is at most .
When a convex lattice polygon has 7, 8, 9 , 11, or 12 interior points, Table 1 of –derived from exhaustively indexing the lattice polygons with interior points–indicates that has maximum lattice width 4, 4, 5, 5, or 6 respectively; from this it follows that the lattice width is at most when is one of these values. ∎
As such, the only lattice polygons that can have an associated curve with no divisors having negative Brill-Noether number are those with ten or fewer than seven interior points. We now show that there is no such curve associated with a lattice polygon with ten interior points.
If is smooth and has genus 10, then there exists a divisor on with negative Brill-Noether number.
We have that the Newton polygon associated to will have ten interior points. As indicated in section 5.9 of , a convex lattice polygon with 10 interior points has (up to lattice transformations) 22 possible interior lattice polygons. With the exception of the polygon displayed in Figure 2, all have lattice width at most 3, and none is equivalent to a lattice polygon that is the convex hull of the points , , and for some , so by Theorem 2.5 unless is the polygon indicated in Figure 2 we have that has lattice width at most 5, so our associated curve will have gonality at most 5 by Lemma 3.2 and therefore some divisor of rank one having a negative Brill-Noether number.
And if is equivalent under some lattice transformation to the polygon in Figure 2, then without loss of generality that the Newton polygon of our is in fact that of Figure 2, so from the proof of Theorem 9 of  we have that is the smooth intersection of two cubic surfaces in , namely and a cubic surface described by . Then will have gonality 6 generally, implying we cannot find a divisor of rank 1 with negative Brill-Noether number. Suppose that we have such a curve with no divisors of rank 1 having negative Brill-Noether number. By the Recognition Theorem of , such a curve will have a Clifford dimension of 3, where the Clifford dimension is the minimal value of such that some divisor of rank has a degree that achieves the minimum value of over all divisors on , the so-called Clifford index. Therefore, we have that its Clifford index is at most 3, since otherwise some divisor of rank 1 would achieve it. So there is some rank-3 divisor of degree at most 9 achieving the Clifford index, and such a divisor has negative Brill-Noether number. So every curve associated with has a divisor with negative Brill-Noether number. ∎
4. Curves of genus at most 6
In this section we establish Theorem 1.2; the partial results of Corollary 3.4 and Propositions 3.5 and 3.6 together establish that no exceptional interior polygon has more than six points, so this process reduces to examining the interior polygons having at most six points.
First, we note that Figure 1 lists up to lattice transformation every interior polygon having six or fewer vertices whose lattice width does not immediately imply the existence of divisors with negative Brill-Noether number in an associated curve, with the exception of the interior polygon in Figure 3, which has associated curves that are plane curves of degree five and hence of Clifford index 1 and Clifford dimension 2. Therefore, any curve having Figure 3 as an interior polygon will have some rank 2 degree 5 divisor, and hence will have some divisor with negative Brill-Noether number.
As such, to establish Theorem 1.2 we now need only show that the 11 polygons in figure 1 have associated curves that have no divisors with negative Brill-Noether number. A critical tool in our analysis will be the Riemann-Roch Theorem, which we state here for completeness.
If is the canonical divisor on a curve of genus , and any divisor on the same curve, then we have
where r(D) is the rank of .
In practice, this means that given a divisor with negative Brill-Noether number, if is sufficiently large then will be another divisor with negative Brill-Noether number but with smaller rank. We make this precise with the following proposition
If a curve of genus has a divisor with negative Brill-Noether number, it has a divisor with negative Brill-Noether number of rank 1. If a curve of genus 5 or 6 has a divisor with negative Brill-Noether number, it has a divisor with negative Brill-Noether number of rank 1 or 2.
We illustrate the proposition in the case of a curve with ; the other cases are analogous. Suppose has a divisor of degree with negative Brill-Noether number and rank . If , we have that subtracting some set of points will produce a divisor of degree at most 4 and rank 2, so will have Brill-Noether number at most . Otherwise, by the Riemann-Roch theorem, we have has rank and has degree , and so has Brill-Noether number . But we observe that has this same Brill-Noether number, so has negative Brill-Noether number and has degree at most 3, so by our previous argument has a divisor of rank at most 2 with negative Brill-Noether number. ∎
Based on this, it suffices to check that the gonality of curves having the Newton polygons in Figure 1 will be at least , and that the minimum degree of a rank two divisor on a curve with Newton polygon among the bottom five in Figure 1 is at least .
But the gonalities for curves associated to all eleven interior polygons are already known. In particular, in the proof of Theorem 4 of  Castryck and Cools determine that the gonality of a certain general family of smooth curves on the toric surface associated their Newton polygon–those which are non-degenerate with respect to their Newton polygon, with interior Newton polygon among those in Figure 1, all have gonality . Hence, curves are associated with all eleven interior polygons that have no rank one divisors with negative Brill-Noether number.
And any curve associated to one of the bottom five interior polygons non-degenerate with respect to its Newton polygon has Clifford index 2 by Theorem 9 of , whence any divisor on of rank 2 has degree at least six and hence has negative Brill-Noether number.
I would like to thank Sam Payne for suggesting the project and for guidance throughout the process. I also would like to thank Dustin Cartwright for many helpful conversations and Dhruv Ranganathan for his comments on drafts of this paper. Finally, I am grateful to the NSF for its support.
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