On the unirationality of moduli spaces of pointed curves

We show that $\mathcal{M}_{g,n}$, the moduli space of smooth curves of genus $g$ together with $n$ marked points, is unirational for $g=12$ and $n\leq 4$ and for $g=13$ and $n \leq 3$, by constructing suitable dominant families of projective curves in $\mathbb{P}^1 \times \mathbb{P}^2$ and $\mathbb{P}^3$ respectively. We also exhibit several new unirationality results for moduli spaces of smooth curves of genus $g$ together with $n$ unordered points, thus showing that they are unirational for $g=10,11$ and $n\leq 7$ or $g=12$ and $n \leq 6$.


Introduction
The geometry of algebraic curves varying in families is a very fascinating and old topic, dating back to the nineteenth century. The interest around this subject naturally led to the definition of the moduli space  of smooth curves of genus over the complex numbers; the study of its birational geometry (or the geometry of its Deligne-Mumford compactification  ) has become a very active research area, especially after the unexpected results of Harris-Mumford-Eisenbud [HM82,EH87]: they showed that  is of general type for ≥ 24, thus contradicting a long-standing conjecture by Severi about its unirationality for any . The unirationality for ≤ 10 being already implied by classical results, a great deal of work has ever since been devoted to the study of the birational geometry of  for the remaining cases, leading to the unirationality for ≤ 14 [Ser81, CR84,Ver05] and the uniruledness for ≤ 16 [CR86,BV05]. The cases  22 and  23 are believed to be of general type, but a full proof is not yet available, at least to our knowledge (see [JP18]); nothing is known for the remaining genera.
More recently, the birational geometry of other moduli spaces has been considered and studied. For instance, one can consider isomorphism classes of curves together with additional structures such as finite maps to ℙ 1 (which leads to Hurwitz spaces, see, e.g., [ST18]), or curves equipped with special line bundles. Being of independent interest, these spaces can also be used to shed further light on the geometry of the underlying spaces  . This paper concerns the study of the birational geometry of two of these spaces, namely the moduli space  , parametrising stable nodal -pointed genus curves, and its quotient  , by the permutation group , which is usually referred to as the universal symmetric product of degree and parametrises smooth curves of genus together with unordered points. Our aim is to provide new unirationality results for some of these spaces in the range 10 ≤ ≤ 13. Being interested only in the birational geometry of these spaces, we will mostly deal with  , and  , instead of their compactifications. By convention, by the Kodaira dimensions of these spaces we will mean the Kodaira dimensions of their compactifications.

On the birational geometry of  ,
The study of the birational properties of the moduli spaces of pointed curves  , can give extra tools for a better understanding of the properties of the spaces  . To highlight this worthiness and mention one application, we remark that the unirationality of  14,2 was a key ingredient for proving that  15 is rationally connected in [BV05].
The moduli spaces of curves  are varieties of general type except for a finite number of cases, occurring for small values of ; the same principle holds true also for  , , at least for > 3. Indeed, on the one hand Logan [Log03] exhibited a natural number ( ) for each fixed genus ≥ 4 such that  , is of general type for = ( ) (and hence, by the subadditivity of Kodaira dimensions, for ≥ ( )). On the other hand, classical constructions of dominant families of curves allow us to easily prove the unirationality for small values of and . Even though a full characterisation of the map is still missing, its behaviour is unknown only for a relatively small number of cases.
For each small genus , we can try to determine when, as increases,  , passes from having negative Kodaira dimension to being of general type. More precisely, we can ask for which values of the moduli space  , is, respectively, unirational, uniruled, of non-negative Kodaira dimension, or of general type. In Table 1 we report on the known results: following an earlier notation, we have indicated for each genus 2 ≤ ≤ 16 the number ( ) (respectively, ( )) such that  , is unirational (respectively, uniruled) for ≤ ( ) (respectively, ≤ ( )). The number ( ) denotes the smallest number of points for which the Kodaira dimension of  , is known to be non-negative.
We do not intend to provide here an accurate account on the several contributions which led to single results in Table 1 We remark that in [Log03] it was previously claimed that  11, is unirational for ≤ 10. However, Barros in [Bar18] noticed that the original argument contained a flaw and only proves the uniruledness of  11, in that range. Barros managed to show that  11, is unirational for ≤ 6, and that it is not unirational for = 9 or = 10.
Thus, one cannot prove the unirationality of  , for up to ( ), as there certainly are cases which are uniruled but not unirational. However, it is not unreasonable to expect that the gap ( ) − ( ) in Table 1 can be reduced in many cases, especially for ≥ 10, where fewer results are known.
The first part of this paper (Section 3) is devoted to narrowing this gap for some cases. Our main result is the following: Theorem 1.1. The space  12, is unirational for ≤ 4; the space  13, is unirational for ≤ 3.
The approach we use is as follows. For a fixed genus ∈ {12, 13}, we produce a unirational family of projective curves of genus dominating  . Then we show that general elements of this family can be linked via hypersurfaces of a suitable degree to particular auxiliary curves. If wisely chosen, these auxiliary curves will be contained in more hypersurfaces of that degree, allowing us to reverse the process and impose a certain number of marked points on the elements of the original unirational family.
We use projective models in ℙ 3 and ℙ 1 × ℙ 2 for  13, ,  12, , respectively. The family of curves of genus 13 is obtained building upon some result proved in the second part of the paper. The construction of the family of curves of genus 12 is based on a particular construction in [Gei13], which we recall in details. Two alternative proofs for the unirationality of  12, for ≤ 3, building upon a different construction or a family constructed in [Ser81], are also provided.

On the birational geometry of  ,
Let us consider the universal symmetric product of degree , constructed as the quotient space  , / and usually denoted by  , . In this paper, we will denote it by  , , to stress out that its elements correspond to the choice of unordered points on a genus curve and to preserve somehow its geometric link with  , .
In a similar fashion for  , in Section 1.1, one can wonder for which pairs ( , ) the space  , is unirational, uniruled, of positive Kodaira dimension, or of general type. On the one hand, in contrast to what happens for ordered points, where the unirationality of  , implies the unirationality of  , −1 , for unordered points we have to face each case separately. On the other hand, unordered points can be easily seen as an effective divisor, allowing us to use a different approach.
12 13 14 15 16 17 18 19 20 21 ( ) 10 11 10 10 9 9 9 7 6 4 Table 2 The unirationality of  , for ≥ 10 and small , however, still remains mysterious, as no positive results are available besides the ones easily deducible from the unirationality of  , . The second part of this paper (Section 4) is devoted to proving several new unirationality results. Our main contribution is the following Theorem 1.3. The space  , is unirational for = 10, 11 and ≤ 7 or for = 12 and ≤ 6.
Except for the case  12,5 , where we adopt an ad hoc construction, the approach we use for the unirationality of  , is as follows. Suppose < . An element of  , is regarded as a pair ( , ), where is a degree line bundle on the curve . The Serre dual divisor of embeds in some projective space. We exhibit dominant unirational constructions for families of such curves for each case of interest. Moreover, we show that general elements of each family can be linked via hypersurfaces of a suitable degree to auxiliary curves. In particular situations we can reverse the process and impose a certain number of additional points on the elements of the original unirational family, so that the unirationality of  , for < ≤ + follows.
Acknowledgements. The authors are grateful to the Max Planck Institute for Mathematics in the Sciences of Leipzig, Germany, where the initial phase of this work was carried out.

Preliminaries
For the sake of clearness, we quickly recall here a few facts about Brill-Noether theory, for which we refer to [ACGH85], and a few facts on liaison theory. is non-negative. In such case, the Brill-Noether scheme

Brill-Noether Theory
has dimension . More generally, one can define the universal Brill-Noether scheme as

Liaison
Definition 2.1. Let and be two curves in a projective variety of dimension with no embedded and no common components, contained in − 1 mutually independent hypersurfaces ⊂ which meet transversally. Let denote the complete intersection curve ∩ . and are said to be geometrically linked via if ∪ = scheme-theoretically.
If and are assumed to be locally complete intersections and to meet only in ordinary double points, then | = ( ∩ ) and the arithmetic genera of the curves are related as follows: The above relation and the equality deg + deg = deg can be used to deduce the genus and degree of from the genus and degree of .
Let = ℙ 1 ×ℙ 2 and be a curve of genus ( ) and bidegree ( 1 , 2 ). Let 1 , 2 be two hypersurfaces of bidegree ( 1 , 1 ) and ( 2 , 2 ) containing and satisfying the above hypotheses. The genus and the bidegree of are For curves embedded in a projective space ℙ , the invariants ( ), of the curve can be computed via where the 's are the degrees of the − 1 hypersurfaces cutting out .

Computational verifications
In this paper we will often need to check on some explicit examples that some open conditions are generically satisfied. In order to do that, we will make use of the software [GS]; the supporting code for this paper has been collected in [KT20]. Although we could a priori run our computations directly on ℚ, this can increase dramatically the required time of execution. Instead, we can work over a finite field , and view our choice of the initial parameters in as the reduction modulo of some choices of parameters in ℤ. The so-obtained example can be seen as the 3 New unirationality results for  12, and  13, In this section we will prove the unirationality of  12, for ≤ 4 and  13, for ≤ 3. The strategy for proving these results is similar: we will exhibit a rational family of projective curves of genus 12 (respectively, 13) which is dominant on the corresponding moduli space  12 (respectively,  13 ). Both these families are constructed via liaison, respectively in ℙ 1 × ℙ 2 and ℙ 3 : they are obtained by linking the desired curve to some auxiliary curve via two hypersurfaces of suitable degrees. As it turns out, the number of such hypersurfaces containing is strictly bigger than 2, allowing us to impose a certain number of points on the desired in a rational way. Parts of the proofs are based on the explicit computation of single examples over a finite field, allowing us to show that some assumptions on the involved geometric objects, which correspond to open conditions, are generically satisfied.

 12,
The key step for proving the unirationality of  12,4 will be the exploitation of a particular case of a construction by Geiß, which can be found in [Gei13] and which we briefly recall in the next section.
In particular, several cases for = 6 and all the cases for = 7 were proved by establishing a correspondence between curves in ℙ 1 × ℙ 2 and certain submodules of the dual of their Hartshorne-Rao modules, in a similar fashion to Chang-Ran's approach in [CR84].
We are mostly concerned with the case ( , ) = (12, 7), as we will use the dominant family exhibited by Geiß to deduce the unirationality of  12,4 . In what follows we briefly recall Geiß' construction for this specific case, in order to better present our argument and to provide some details which were omitted in [Gei13]. The interested reader can find another specific case, namely ( , ) = (10, 6), examined in details in [CHGS12, Appendix A].
Let → ℙ 1 be a general element in  12,7 . We consider a line bundle on of degree 10 such that the map given by it and the assigned 1 7 embeds in ℙ 1 × ℙ 2 as a curve of bidegree (7, 10) of maximal rank; the existence of such line bundle corresponds to some open conditions on  12,7 , which can be seen to be satisfied for a general element through the realisation of an explicit example. A Hilbert function computation shows that the truncated ideal ∶= ( ) ≥(4,3) admits a minimal free bigraded resolution over the Cox ring of ℙ 1 × ℙ 2 of the form Let us denote by the terms in the above resolution, e.g., 1 = (−4, −4) 9 ⊕ (−4, −3) 3 . Then ∶= coker ∨ is a module of finite length, called the truncated deficiency module.

The first terms of a minimal free resolution of look like
one can prove that composing with a general map ∨ 1 → we obtain a matrix whose kernel is isomorphic to . The entries of the corresponding induced map → ∨ 1 generate , allowing us to recover and thus the original curve from its truncated deficiency module .
This correspondence can be exploited by constructing a rational family of modules and showing that such family leads to a family of curves in ℙ 1 × ℙ 2 which is dominant (by considering the first projection) on  12,7 . As it turns out, the main difficulty lies in the construction of , as a general matrix ∨ 2 → ∨ 3 will produce a module of finite length which in general has Hilbert function different from the one is expected to have. In our specific case, will have dimension zero in all bidegrees but the cokernel of a general matrix (5, 4) 10 ⊕ (4, 5) 7 → (5, 5) 6 , however, will be zero-dimensional in bidegree (−4, −4). It is thus necessary to construct -in a rational way -a family of matrices whose cokernels have Hilbert function as in (3.1). Following Geiß' construction and by using Macaulay's inverse systems, we implemented in [KT20] the construction of a family of curves of genus 12 and bidegree (7, 10) in ℙ 1 × ℙ 2 , dominant on  12 .
We now focus on genus 12 curves. As explained in the previous section, we have a unirational family of curves of genus 12 and bidegree (7, 10) in ℙ 1 × ℙ 2 which is dominant on  12 . By (3.2), a curve of this family lies on at least 2 independent hypersurfaces of bidegree (2, 4). We can therefore consider the complete intersection curve cut out by two such hypersurfaces and link to a curve . If we assume that and meet transversally and that is smooth, by (2.2) will have genus 4 and bidegree (9, 6).
By (3.2), a general curve with the same genus and bidegree of will be contained in at least 6 independent hypersurfaces of bidegree (2, 4). This means that we can perform liaison back: via two general hypersurfaces of bidegree (2, 4) containing it, we can link to a curve , which will be again of genus 12 and bidegree (7, 10) in general. Moreover, since we started from a unirational dominant family on  12 , and since the choice of the above hypersurfaces is rational, by performing liaison forth and back we obtain again a unirational dominant family.
The fact that is contained in at least 6 hypersurfaces of bidegree (2, 4) allows us to impose the choice of 4 general points on the so-constructed , obtained via liaison by choosing the hypersurfaces passing through those points. Since the choice of points in ℙ 1 × ℙ 2 is rational, we obtain a unirational dominant family of curves of genus 12 together with (up to) 4 marked points.
We can thus prove the following Theorem 3.3. The moduli space  12, is unirational for ≤ 4.
Proof. The only thing left to show is that, for a general curve in the family constructed above, the residual curve obtained via liaison is indeed smooth and intersects transversally. These are open conditions on the family, and can be checked through the realisation of a specific example, as we do in [KT20].

Another proof for the unirationality of  12,3
We present here another argument which yields the unirationality of  12, for ≤ 3. Even if superseded by Theorem 3.3, we believe it is of independent interest as it relies only on geometric arguments and does not depend on Geiß' construction and the homological approach used in it. As mentioned in Remark 3.6, one could get a different proof by using a family of curves constructed by Sernesi, obtained again via a homological approach.
By Riemann-Roch, the Serre dual bundle ⊗  (−1) has a 5-dimensional space of global sections and degree 15. Hence, it is expected to embed the curve in ℙ 4 as a curve of degree 15. In this embedding, is contained in at least 4 cubic hypersurfaces. Let ⊂ ℙ 4 be the curve linked to via the complete intersection of three such cubic hypersurfaces. By (2.3), is a curve of genus 9 and degree 12. Again by Riemann-Roch we have h 0 ( ⊗  (−1)) = 1: this means that the Serre dual divisor corresponds to an element of  0 9,4 , or to the class of an effective divisor of degree 4 on .
Proof. We claim that the above construction can be reversed, i.e, that there exists a chain of correspondences These maps should be actually thought of as maps between some components of the spaces here above; maps labelled by correspond to considering the Serre dual model, while maps labelled by correspond to taking suitable linkages. The reversibility comes from the fact that all the open assumptions we made about the generality and smoothness of the objects involved and the transversality of the curves in liaison hold true in some specific examples, as verified in [KT20].
What remains to show is that this construction provides a unirational family which is dominant on  12,3 . With the above argument, the first liaison on the left produces a curve of genus 12 and degree 12 in ℙ 3 from a curve of genus 15 and degree 13 and the choice of 2 quintic hypersurfaces containing it. lies on at least 5 independent quintic hypersurfaces; therefore, for the choice of at most three general points in ℙ 3 , there are at least two such quintics passing through the points. The corresponding complete intersection will link to with up to three marked points.
By Riemann-Roch, the known unirationality of  9,4 (Section 1.1) implies the unirationality of  0 9,4 . Following (3.5) backwards, we find a unirational family which is dominant on some component of  3 12,12 . The conclusion follows as soon as we show that the so-constructed curves lie in the irreducible component of  3 12,12 which dominates  12 : for this sake, it is sufficient to show that in general the Petri map H 0 ( (−1)) ⊗ H 0 ( (1)) → H 0 ( ) is injective. This condition is open and has been checked for specific examples in [KT20] by showing that there are no linear relations among the generators of Γ * ( ) in degree −1.
Remark 3.6. In [Ser81], Sernesi constructed a rational family of curves of genus 12 and degree 12 in ℙ 3 which is dominant on  12 , proving thus the unirationality of  12 . Another way to prove Theorem 3.4 is to use this family and perform a liaison with respect to 2 quintics in ℙ 3 forth and back. Once again, the choice of hypersurfaces of a given degree containing a given curve is rational and a general curve of genus 15 and degree 13 in ℙ 3 lies on exactly 5 quintics, allowing one to impose up to 3 points on a general genus 12 curve.

 13,
Let be a general curve of genus 13. Since (13, 3, 13) = 1, admits a 3 13 , which gives an embedding of as a curve of degree 13 in ℙ 3 . Since  (5) is non-special by Riemann-Roch, is contained in at least 3 = 8 5 − (5 ⋅ 13 + 1 − 13) quintic hypersurfaces. The complete intersection cut out by two such hypersurfaces links to a curve , which by (2.3) has genus 10 and degree 12 provided that it is smooth and that intersects transversally.
Let be the divisor associated to the embedding of ⊂ ℙ 3 . Riemann-Roch yields h 0 ( − ) = 1, and therefore the Serre dual divisor determines an element of  0 10,6 , whose general element can be interpreted as a curve together with 6 unordered points on it.
Proof. The above construction can be reversed: indeed, a curve of genus 10 and degree 12 in ℙ 3 lies on at least 5 independent quintic hypersurfaces. For the choice of ≤ 3 general points in ℙ 3 , there are at least 2 such hypersurfaces passing through the points so that we can reverse the liaison construction to obtain a new genus 13 curves together with up to 3 marked points. The conclusion then follows as soon as we prove that  0 10,6 is unirational and that the unirational family provided by the above construction dominates  13 .
A general element in  0 10,6 gives 6 unordered points on ; conversely, by Riemann-Roch 6 general points on a general curve of genus 10 provide an element of  0 10,6 . On the one hand,  10,6 is not known to be unirational (Section 1.1); on the other hand, we only need 6 unordered points, as we want to consider them as a divisor on the curve. Hence, it suffices to prove the unirationality of  10,6 , a result provided by Theorem 4.1 below.
To prove that the so-constructed unirational family of curves of genus 13 and degree 13 dominates  13 , it is sufficient to show that in general the Petri map H 0 ( (−1)) ⊗ H 0 ( (1)) → H 0 ( ) is injective. This open condition has been checked for one particular example in [KT20] by showing that there are no linear relations among the generators of Γ * ( ) in degree −1.

New unirationality results for  ,
The aim of this section is to prove the following  This, together with the known results reported in Section 1.1 and Theorem 3.3, yields Theorem 1.3.
Proof. Let us prove the unirationality of  12,5 first. Let us consider 5 general points on a general curve of genus 12. The Serre dual divisor corresponding to gives an embedding of as a curve of degree 17 in ℙ 6 . The number of quadrics of ℙ 6 containing is at least 5, so can be linked to a curve via a complete intersection of type (2 5 ). If we assume that is smooth and meets transversally, will have degree 15 and genus 9 by (2.3); moreover, will be contained in at least 6 quadrics. By [Ver05, Theorem 1.2], the unique irreducible component of the Hilbert scheme of curves of genus 9 and degree 15 in ℙ 6 which dominates  9 is unirational. Since  6 12,17 is irreducible, the conclusion then follows if we show that, for a general in such component, the liaison works as expected and the curve obtained by reversing the liaison construction is non-degenerate. This condition and the assumptions we made on above are open in moduli and can be checked through the realisation of particular examples, as we do in [KT20].
For the remaining cases in (4.2) we adopt a common approach, an instance of which can be found in Example 4.3 below; the general approach differs only for the numerology involved and can be explained as follows. Assume that  , is unirational and consider < − 3 general points on a general curve of genus , so that the Serre dual divisor of the points leads to a general element of  − −1 ,2 −2− . This dual divisor embeds as a curve of degree in a suitable projective space ℙ , where = − − 1. We then search for possible reversible liaison constructions and try to impose the choice of further points, in order to obtain a projective curve with marked points on it. The projective model yields by construction unordered points, which together with the additional provides the unirationality for  , + .
In Table 3 we collect for each of the cases of interest the values for , , , , as well as the liaison type L we use, which will be always given by −1 hypersurfaces of the same degree ℎ. After choosing ℎ, the genus and degree of the curve obtained via liaison are fixed. As it turns out, with these data we can impose only = 1 further point on genus curves, hence = + 1 in these cases.
L 10 5 13 4 3 3 12 14 6 10 6 12 3 5 2 13 13 7 11 6 14 4 3 3 9 13 7 12 5 17 6 2 5 9 15 6 Theorem 4.1 is proved as soon as we show that the assumptions we make on the occurring linkages (namely, the smoothness of and the transversality of the intersections of and ) are satisfied in general. These correspond to open conditions, which are then proved to be verified for general elements of the spaces under investigation via the construction of specific examples, done in [KT20].
Example 4.3. For the unirationality of  10,6 the aforementioned approach goes as follows. We can start from the unirationality of  10,5 granted by Table 1, so that = 5. A general element of  10,5 provides a general curve of genus = 10 and degree = 2⋅10−2−5 = 13 in ℙ =10−5−1=4 . Such curve is contained in at least 5 cubic hypersurfaces, so we can consider linked to via (3 3 ); if we assume that is smooth and meets transversally, then by (2.3) will have genus = 12 and degree = 14.
is contained in at least 4 cubic hypersurfaces, so that we can reverse the construction and impose = 4 − ( − 1) = 1 further point on a general curve (which will be a priori different from the one we started from). The unirationality of  10, + =6 follows.