Gaussian maps for singular curves on Enriques surfaces

We give obstructions - in terms of Gaussian maps - for a marked Prym curve $(C,\alpha,T_d)$ to admit a singular model lying on an Enriques surface with only one $d$-ordinary point singularity and in such a way that $T_d$ corresponds to the divisor over the singular point.


Introduction
The article deals with the problem of finding obstructions -in terms of Gaussian maps -for a Prym curve (C, α) to admit a singular model (with prescribed singularities) in a polarized Enriques surface (S, H). Let us briefly introduce the setting. Let X be a smooth complex projective variety, and let L and M be two invertible sheaves on X. If L = M one usually writes Φ 1 X,L and since it vanishes on symmetric tensors, one usually considers its restriction to 2 L. Gaussian maps were introduced by Wahl who showed, in [43], that if (S ′ , H ′ ) is a polarized K3 surface and and C ′ ∈ |H ′ | is a curve then the Gaussian map (also called Wahl map) Φ 1 ω C is not surjective (see also [7] for a different proof). On the other hand Ciliberto, Harris and Miranda proved in [11] that the Wahl map is surjective as soon as it is numerically possible, i.e. for g ≥ 10, g = 11. The converse also holds. In [1] Arbarello, Bruno and Sernesi proved that a Brill-Noether-Petri general curve with non-surjective Gaussian map lies in a K3 surface or on a limit thereof. See also [10] for a result relating the corank of the Gaussian map and r-extendibility. Analogous problems for Enriques surfaces have also been studied by some authors. Indeed, let (S, H) be a polarized Enriques surface and let C ∈ |H| be a smooth curve and α := K | C . In [4] it is proven that the Gaussian map Φ ω C ,ω C ⊗α is not surjective, whereas in [15] it is shown that for the general prym curve (C, α) of genus g ≥ 12, g = 13, 19 the map is surjective. Gaussian maps have been studied and used by many authors, either in relation to extendibility questions, we mention e.g. [4], [5], [6] - [24], [8] - [16] - [40], [9], [23], [31] - [32], [20] (see also [34] for a complete survey), or in relation to the second fundamental form of Torelli-type immersions, e.g. [17], [18], [19], [21]. [25]. The result of Wahl was generalized by some authors, e.g. Zak-L'Vovsky, who proved the following theorem, that we are going to use.
Theorem 0.1 ( [33]). Let C be a smooth curve of genus g > 0 and let A be a very ample line bundle on C, embedding C in P n for n ≥ 3. If C ⊂ P n is scheme-theoretically a hyperplane section of a smooth surface X ⊂ P n+1 , then the Gaussian map Φ ω C ,A is not surjective.
Similar questions for singular curves on K3 surfaces are discussed and solved by Kemeny in [28]. In the article the author asks whether one can give an obstruction in terms of suitable gaussian maps for a curve to have a nodal model lying on a K3 surface. Following the author notations, denote bȳ M h,2l the moduli stack of smooth curve of genus g with 2l marked point and by M h,2l =M h,2l /S 2n the stack of curves with unordered marking. Let h, l be two positive integers and [C, T ] ∈ M h,2l . The author introduces the marked Wahl map: (0.1) W C,T : 2 H 0 (C, K C (−T )) → H 0 (C, K 3 C (−2T )). Then the following theorems are proven. 1 Theorem 0.2 ( [28]). Fix any integer integer l ∈ Z. Then there exists infinitely many integers h(l), such that the general marked [(C, T )] ∈ M h(l),2l has surjective marked Wahl map. Now denote by V n g,k the stack parametrizing morphisms [(f : C → X, L)] where (X, L) is a polarized K3 surface with L 2 = 2g−2, C is a smooth connected curve of genus p(g, k)−n with p(g, k) := k 2 (g−1)+1, f is birational onto its image and f (C) ∈ |kL| is nodal. 28]). Assume g − n ≥ 13 for k = 1 or g ≥ 8 for k > 1, and let n ≤ p(g,k)−2

5
. Then there is an irreducible component I 0 ⊆ V n g,k such that for a general [(f : C → X, L)] ∈ I 0 the marked Wahl map W C,T is non surjective, where T ⊆ C is the divisor over the nodes of f (C).
The same marked Wahl maps have been studied by Fontanari and Sernesi in [35], where they proved, using very different methods from [28], the following theorem.
Theorem 0.4 ( [35]). Fix an integer g ≥ 9 Let (S, H) be a polarized K3 surface with P ic(S) = ZH and H 2 = 2g − 2. Let C be a smooth curve of genus g − 1 endowed with a morphism f : C → S birational onto its image and such that f (C) ∈ |H|. If T = P + Q ⊆ C is the divisor over the singular point of f (C), then the Gaussian map Φ ω C −T,ω C −T is not surjective.
Our paper deals with a similar problem for singular curves on Enriques surfaces. Let (S, H) be a polarized Enriques surface and C a smooth curve having a morphism f : C → S birational onto its image and such that f (C) ∈ |H| has exactly one ordinary d-point or a cusp, in case d = 2. Denote by T d the divisor over the singular point and set α = f * K S . Then (C, α, T d ) is a marked Prym curve. We investigate the behaviour of the following mixed Gaussian-Prym maps: . More precisely we have the following.
Theorem 0.5. Let (S, H) be a polarized Enriques surface with H 2 = 2g − 2 and let d ≥ 2. Suppose that either (i) S is a very general Enriques surface and ϕ(H) ≥ √ 2(d + 2), or (ii) S in unnodal and ϕ(H) ≥ 2(d + 1). Set g ′ = g − d 2 and let C be a smooth curve of genus g ′ having a birational morphism f : C → S onto its image and such that f (C) ∈ |H|, f (C) has exactly one ordinary d−point or a cusp, in case d = 2.
Set α = f * K S | C and let T d = p 1 + ... + p d be the divisor over the singular point. Then the Gaussian maps Φ ω C ,ω C −T d +α and Φ ω C −T d ,ω C −T d +α are not surjective.
Here, with "general", we mean in a non empty Zariski-open subset of the moduli, with "very general" we mean outside a countable union of proper Zariski-closed subset. The proof is along the same lines of Theorem 0. 4 On the contrary, when one considers a general marked Prym curve the aforementioned maps are "tendendially" surjective. Indeed, let S be the following set: and denote by R g,d the moduli space of d−marked Prym curve. We prove the following.
Theorem 0.6. Let (g 1 , d 1 , d 2 ) be in S (5.2), and g = ( is a general element in R g,d , then the Gaussian maps In case d = 2, 3 or d = 4 (see example 1 ) we obtain the surjectivity for all genera greater than or equal to 41. More generally, for every m we obtain infinitely many genera for which the marked Gaussian maps (we are considering) are surjective. We expect our result far from being sharp (see remark 5.3).
We briefly explain how the paper is organized. In section 1 we recall the definition of the Gaussian maps and prove (proposition 1.2) a slight modification of [35], Theorem 8 (see also Theorem 9). This is a result relating cokernels of Gaussian maps in different embeddings. In section 2 we prove theorem 0.5, following the same strategy of the proof of theorem 0.4 ([35]). In particular, we show some results regarding very ampleness of line bundles on the blow-up of Enriques surfaces. In 3 we prove the surjectivity of the marked Prym-gaussian maps for a certain class of d− marked Prym curves living in the product of two curves. In section 4 we give a lower bound for the gonality of curves living in the product of a curve with P 1 (see proposition 4.1 ) and we prove a lemma about very ample line bundles on curves. We will use them in the proof of theorem 0.6. In section 5 we finally prove theorem 0.6.

Acknowledgments
I deeply thank Paola Frediani and Andreas Leopold Knutsen for several discussions about the paper and for their fundamental remarks and suggestions. I also want to thank Andrea Bruno, Margherita Lelli Chiesa, Angelo Felice Lopez, and Edoardo Sernesi for a conversation about the paper and their suggestions.

Cokernels of Wahl maps
We briefly recall the definition of the Gaussian maps, and their different interpretations, which will be used in the sequel. See for example [41] or [42] for the details. Let X be a smooth complex algebraic variety. Let L and M be two line bundles on X. Let q i : X × X → X, i = 1, 2 be the two projections. Consider the short exact sequence defining the square of the diagonal ∆ X×X and tensor it with q * The first Gaussian map associated with L and M is defined as the map induced at the level of global sections: and tensor it with O P r (1): and so we conclude. Now consider a twist by L ⊗ M of the conormal exact sequence: One can show that under the aforementioned identification, i.e. Φ L,M is the map induced at the level of global section in 1.2. Now we recall a very useful construction of Lazarsfeld. ). Let p 1 , ..., p n ∈ X be distinct points such that L(− n i=1 p i ) is generated by global sections, and h 1 (L(− n i=1 p i )) = h 1 (L). Then one has an exact sequence: We now observe that a slight modification of [35],Theorem 8, gives the following result which relates cokernels of gaussian maps in different embeddings. In the following X = C will be a smooth complex algebraic curve.
Proposition 1.2. Let C be a smooth complex projective algebraic curve. Let T n = p 1 + ... + p n be an effective divisor of degree n on C with p i = p j for i = j. Let L and M be two very ample line bundles such that L − T n is very ample and h 1 (L) = h 1 (L − T n ). Then there exists a surjection between the cokernels of the gaussian maps: The proof follows the same steps of [35],Theorem 8. We present it for completeness.
Proof. Consider the following commutative diagram. The first two rows are (1.2) for the line bundles L and L − T n , the second column is (1.3) twisted by M , and the third column is just the restriction modulo the identification Passing to cohomology we obtain . Then ker(H 1 (g)) = 0.

Non surjectivity
In this section we are going to prove theorem 0.5. We proceed in a similar way as in [35]: we will obtain the non-surjectivity result applying theorem (0.1) and a result about very ampleness of line bundles on the blow-up of Enriques surfaces.
Let S be an Enriques surface. First we recall the definition of two positivity measures: the ϕ−function and the Seshadri constant of a big and nef line bundle H on S. The first one is defined as The following holds: For background and proofs see for example [22]. Now let σ : S ′ → S be the blow-up in a point p. We will now give, in terms of ϕ, sufficient conditions for a line bundle of the form σ * H − lE to be big and nef.
In the following, when we say a "very general" Enriques surface, we mean that as a point in the moduli space of Enriques surfaces, it lives outside a countable union Zariski-closed subsets. We also recall that a nodal Enriques surface is one which contains −2 curves. In the moduli space of Enriques surface theese correspond to a divisor. An Enriques surface not containing a −2 curve is usually called unnodal.
Proposition 2.1. Let S be an Enriques surface and and l ≥ 1 be an integer. Let H be a big and nef line bundle on S and suppose one of the following holds: i) S is a very general Enriques surface, ϕ(H) = l and H is not of the type H ≡ l ii) S is a very general Enriques surface and ϕ(H) ≥ l + 1. iii) S is unnodal and ϕ(H) ≥ 2l. . Then σ * H − lE is big and nef.
Proof. First we show that σ * H − lE is nef. From [38], proposition 5.1.5., it follows that σ * (H) − lE is nef if and only if ǫ(H) ≥ l. In [27], Theorem 1.3, it is shown that if S is a very general Enriques surface then ϕ(H) = ǫ(H). Then, in case i) or ii) we conclude. Now suppose we are in situation iii). From [27], Corollary 4.5, it follows that ǫ(H) ≥ 1 2 ϕ(H) ≥ l and we immediately conclude.
From 2.1 and the hypothesis l ≥ 1 in case ii and iii) we get H 2 ≥ ϕ(H) 2 > 0 and hence σ * H − lE is also big. Consider now the situation i) and suppose that σ * H − lE is not big, i.e. H 2 = l 2 . Then, again by 2.1, we have H 2 = ϕ 2 = l 2 . By [30], Proposion 1.4, we must have H ≡ l( The proof of the next result is a direct application of Reider's Theorem (see [44], Theorem 1, or [22], Theorem 2.4.5).
Proposition 2.2. Let l ≥ 1 and let (S, H) be a polarized Enriques surface. Suppose that either (i) S is a very general Enriques surface, l ≥ 1 and ϕ(H) ≥ √ 2(l + 2), or (ii) S in unnodal, l = 1 and ϕ(H) ≥ 3 √ 2, or l ≥ 2 and ϕ(H) ≥ 2(l + 1). Let σ : S ′ → S be the blow-up in a point and E be the exceptional divisor. Then σ * H − lE is a very ample line bundle on S ′ .
Observe that is also effective. Indeed suppose by contradiction it is not. Then, by Riemann-Roch and Serre duality, where the latter is just the fact that H is ample and L effective. Then we conclude that σ * H ′ −(l+1)E is effective. Now suppose by contradiction that σ * H − lE is not very ample. Since σ * H ′ − (l + 1)E is an effective, big and nef divisor and H 2 ≥ ϕ 2 (H) ≥ 9 + (l + 1) 2 in both cases (i) and (ii), we can apply Reider's theorem. Then there exists a non trivial effective divisor D in S ′ such that either one of the following holds: . Now we show that none of these can happen.
Let D ∼ σ * L − aE, for some L ∈ Pic(S) and a ∈ Z. Suppose we are in case (a). Then we have H ′ L ≤ (l + 1)a + 2 and L 2 = a 2 and so we obtain the following inequalities: where the second inequality follows by Hodge index theorem. If |a| ≥ 2 we obtain which contradicts the hypothesis. If |a| = 1 from 2.2 we get ϕ(H) = ϕ(H ′ ) ≤ (l + 1) + 2 which again is not possible. If a = 0 we get D = σ * L with L effective, not numerically trivial and such that L 2 = 0 and H ′ L ≤ 2. This gives ϕ(H) ≤ 2 and we conclude. Suppose now we are in case (b). As before one have If |a| ≥ 2 we find ϕ(H ′ ) < √ 2(l + 2). If a = 1 then L is an effective divisor such that L 2 = 0 and H ′ L ≤ l + 2. Moreover observe that L is not numerically trivial since otherwise D ≡ −E, which is not possibile because D is an effective non trivial divisor. Therefore we obtain ϕ(H ′ ) ≤ l + 2. a = −1 cannot happen if l ≥ 1 because H ′ is nef and L is effective and L · H ′ = −l. If a = 0 then L 2 = −1. This is not possibile for Enriques surfaces.
Suppose now we are in case (c). Then H ′ L = a(l + 1) and L 2 = a 2 − 2. Then, as before, Observe that if |a| ≥ 2 this gives ϕ(H ′ ) ≤ √ 2(l + 1) and hence we conclude. Observe that |a| = 1 cannot happen because otherwise L 2 = −1 and this, again, is not possible. If a = 0 then L is a effective divisor such that L 2 = −2 and H ′ L = 0. This cannot happen because H ′ · L = (H + K S ) · L, H is ample and L is effective.
We observe that proposition 2.2 has the following corollary.
Consider then the following commutative diagram:

Surjectivity for special curves
We start this section with a proposition giving sufficient conditions for the surjectivity of mixed Gaussian maps for surfaces which are the product of two curves. The idea of computing the rank of gaussian maps on the product of two curves with gaussian maps on the two factors dates back to Wahl ([42], Lemma 4.12). See also , Theorem 3.1) for the second Wahl map.
Then Φ X,L,M is surjective. Proof. We want to relate the gaussian map Φ X,L,M with gaussian maps on C i , i = 1, 2. Let q i : X × X → X, i = 1, 2 the two projections. Recall that Φ X,L,M is given by: 2 the projections and analougly q i,2 : C 2 × C 2 → C 2 . Let (ϕ 1 , ϕ 2 ) the isomorphism which exchange factors: , are the inverse image ideal sheaves or equivalently the pullbacks sheaf (because projections are flat). Now consider the isomorphism of O X -modules: can be read as So we obtain the following commutative diagram:

Taking global section we obtain
In order to show that Φ X,L,M is surjective we will show the surjectivity of ψ. Cleary ψ is surjective if each of the direct sum map is surjective: Let us deal with the first map. The same reasoning will apply also to the second one. Observe that Then we can write And so we obtain Now using that X × X ≃ − → (C 1 × C 1 ) × (C 2 × C 2 ) and Künneth formula we get: ). Under these identifications ψ 1 becomes: and it is given by the tensor product Φ C 1 , Now observe that if deg(L i ), deg(M i ) ≥ 3g i + 2 for i = 1, 2, then by Theorem 1.1 of [3], each gaussian map is surjective and so ψ is.
Remark 3.2. Let X 1 and X 2 be two smooth varieties of any dimension. Let L 1 , M 1 and L 2 , M 2 be two line bundles on X 1 and X 2 respectively. Denote by L = L 1 ⊠ L 2 and M = M 1 ⊠ M 2 . We observe that a similar proof gives a lifting of Φ L,M by Φ X 1 , We are now going to prove a surjectivity result for mixed gaussian maps on curves living in the product of two curves.
Proof. Consider the following commutative diagram Observe that the vertical arrow and π 1 are restriction maps, whereas π 2 comes from the conormal bundle sequence . We prove that Φ L,M , π 1 , and π 2 are surjective. From this we obtain the desired surjectivity result. The surjectivity of Φ L,M is just 3.1. The surjectivity of π 1 will follow from the vanishing of H 1 (X, Ω X ⊗L⊗M (−C)) ≃ H 1 (X, p * 1 ω C 1 ⊗L⊗M (−C))⊕H 1 (X, p * 1 ω C 2 ⊗L⊗M (−C)). Consider the first piece. Observe that
Main construction 3.4. In this remark we consider a construction we will use in the following corollary. First observe that if S is a smooth surface, H is an ample divisor on S and C ∈ |H| is a smooth curve, then the restriction map Pic 0 S → Pic 0 C is injective by Lefshetz hyperlane theorem (see for example [26], theorem C). Now let C 1 and C 2 be two curves. Let X be the product C 1 × C 2 . Let p i : X → C i , i = 1, 2 be the two projections and let D i be effective divisors of degree d i such that |p * 1 D 1 + p * 2 D 2 |. is base point free. Let C be a smooth irreducible curve in the linear system |p * 1 D 1 + p * 2 D 2 |. Let α ′ ∈ Pic 0 (C 1 ) a non trivial 2-torsion element( in particular g(C 1 ) ≥ 1 ). Then α 1 := p * 1 α ′ is a non trivial 2-torsion element in Pic(X) and α := α 1 | C is a non trivial 2-torsion element in Pic(C). Let supp(D 1 ) = {p 1,1 , ..., p 1,d 1 } and denote by T d 2 the divisor of the d 2 intersection points between the fiber p −1 1 (p 1,1 ) and C. Remark 3.5. We observe that a sufficient condition for O X (p * 1 D 1 + p * 2 D 2 ) to be base-point free is that both O C 1 (D 1 ) and O C 2 (D 2 ) are. Observe that if C is any curve of genus g ≥ 1, a general effective divisor D of degree d ≥ g + 1 is basepoint-free. This follows from classical results but we recall it.
Since every divisor of degree 2g is base-point-free, we can restrict to the case g ≥ 2 and g + 1 ≤ d ≤ 2g − 1. Consider first the case d = 2g − 1. Let D ′ be a general divisor of degree 2g − 2 and p ∈ C be a point. Then, by Riemann-Roch, it immediately follows that D ′ + p is a base-point free divisor of degree 2g − 1. Now suppose g + 1 ≤ d ≤ 2g − 2 and consider the Brill Noether variety the last one is birational to Sym 2g−1−d C and hence has dimension 2g − 1 − d. Then the image of 3.5 has dimension 2g − d. On the other hand W d−g d has dimension greater than or equal to ρ(g, d − g, d) = g. We conclude that if d ≥ g + 1 the image of 3.5 is proper subvariety and hence that the general element is base-point-free. Corollary 3.6. Using the construction 3.4 suppose that one of the following holds: and Denote by l i , m i , i = 1, 2 and l ′ i , m ′ i , i = 1, 2 their degrees. To prove the surjectivity of the gaussian maps we want to apply proposition 3.3 with L i , M i , i = 1, 2 in the first case, and L ′ i , M ′ i , i = 1, 2, in the second. Since l ′ i ≥ l i , i = 1, 2, m ′ i ≥ m i , i = 1, 2, it is enough to verify the hypothesis of proposition 3.3 in the first situation. It is easy to see that the conditions become: d 1 ≥ 5, d 2 ≥ 4, d 1 ≥ g 1 + 5, d 2 ≥ g 2 + 4 and d 2 (g 1 − 2) + d 1 (g 2 − 1) > 0. Then we conclude as in the statement.
We end this section with a surjectivity result for the related multiplication maps.
Proposition 3.7. Using the construction 3.4 suppose that d 2 ≥ 3 and d 1 ≥ 4, g 1 ≥ 1, or g 1 = 1 and d 2 ≥ 3. Then where p is the restriction map. In order to prove the surjectivity result, again, it is sufficient to prove that Φ 0 L,M and p are surjective. The multiplication map: decomposes, using the identifications in 3.1 with L 1 = ω C 1 + D 1 − p 1,1 and L 2 = ω C 2 + D 2 , M 1 = ω C 1 + D 1 − p 1,1 + α 1 , M 2 = ω C 2 + D 2 , and Künneth theorem as before, as the tensor product of the multiplication maps on the curves C i : i = 1, 2: Since l i , m i ≥ 2g i + 1, i = 1, 2, each of the multiplication maps is surjective by a classical result of Mumford. The surjectivity of p will follow from the vanishing of H 1 (X, L ⊗ M − C). By Künneth this is isomorphic to Now observe that h 1 (C 2 , L 2 ⊗ M 2 (−D 2 )) = h 1 (C 1 , L 1 ⊗ M 1 (−D 1 )) = 0. This is a consequence of Serre duality and the fact that l i + m i > 2g i − 2 + d i . This ends the proof of the surjectivity of 3.6. An identical proof, with gives the surjectivity of 3.7.

Some useful lemmas
In this section we prove some results we will need in the next one. Let C be a curve. We will need an upper bound on the gonality of curves in the surface C × P 1 , where C is a curve. The proof is very much inspired by [36] (see Lemma 2.8 and Theorem 6.1).
Let p 1 : C × P 1 → C, p 2 : C × P 1 → P 1 be the two projections. Let C 0 be the class of a fiber of p 2 . Recall that P ic(C × P 1 ) = p * 1 (P ic(C)) ⊕ ZC 0 , and that the Néron-Severi is generated by C 0 (class of a fiber of p 2 ) and the class of a fiber of p 1 , which we will call f . We are going to prove the following: • If C is any curve, g(X) > 0 and d 2 ≥ d 1 For the proof we will use the following theorem of Serrano (see [45]): Theorem 4.2. Let X be a smooth curve on a smooth surface S. Let ϕ : X → P 1 be a surjective morphism of degree d. Suppose that either (a) X 2 > (d + 1) 2 , or (b) X 2 > 1 2 (d + 2) 2 and K S is numerically even. Then there exists a morphism ψ : S → P 1 such that ψ | X = ϕ.
Recall that a divisor D is called numerically even if D · E is even for any other divisor E. In our case, being K C×P 1 ≡ −2C 0 + (2g(C) − 2)f , we have that K C×P 1 is numerically even. Before giving the proof of proposition 4.1 we will need the folllowing: Lemma 4.3. Let X ∈ |p * 1 (D 1 ) + d 2 C 0 | be a curve in C × P 1 . Let ϕ : X → P 1 be a morphism such that there exists ψ : S → P 1 such thath ψ | X = ϕ. Then deg(ϕ) ≥ min(d 2 gon(C), d 1 ).
Proof. Let D be a fiber of ψ. Then D ∼ p * 1 B + aC 0 , with a ∈ Z and B a divisor in C of degree b. Numerically: D ≡ bf + aC 0 . From f · D ≥ 0, C 0 · D ≥ 0, and D 2 = 0 one finds a ≥ 0, b ≥ 0 and 2ab = 0. Then we have two cases: (i) a = 0. In this case D ∼ p * 1 B. Then deg(ϕ) = deg(ψ | X ) = X · D = d 2 b ≥ d 2 gon(C), where the latter inequality follows from the observation that the restriction of ψ to a fiber of p 2 gives a morphism C → P 1 of degree greater or equal than C 0 · D = b. And so b ≥ gon(C). (ii) b = 0. In this situation D ∼ aC 0 and then deg(ϕ) = deg(ψ | X ) = aC 0 · X = ad 1 ≥ d 1 .
Since we want this lemma to hold for any effective divisor T m of degree m, we have to suppose m ≤ g − 3. This condition in fact guarantees that h 0 (C, ω c − T m + α) ≥ 2. (a) Suppose ω C − T m + α is not base point free. Then (i) h 0 (C, T m + α) = 0 and there exist a point p such that dim(|2(T m + p)|) ≥ 1; or (ii) h 0 (C, T m + α) ≥ 1 and there exist a point p such that dim(|T m + α + p)|) ≥ 1. (b) Suppose ω C − T m + α is not very ample. Then (i) there exist points p and q such that h 0 (C, T m + α + p) = 0 and dim(|2(T m + p + q)|) ≥ 1; or (ii) there exist points p and q such that h 0 (C, T m + α + p) ≥ 1 and dim(|T m + α + p + q)|) ≥ 1.

Proof.
(a) Suppose ω C − T m + α has a base point p. Then, by Riemann-Roch, Then there exists an effective divisor E such that E ∼ T m + p + α. This gives 2E ∼ 2(T m + p). Now observe that h 0 (2(T m + p) ≥ 2 since otherwise 2E = 2(T m + p) and hence E = T m + p which gives α = 0. This cannot be the case because α is not trivial by hypothesis. (b) Suppose there exists two points p and q such that q is a base point of ω C − T m + α − p. Then, by Riemann-Roch, h 0 (T m + p + q + α) = h 0 (T m + p + α) + 1. If h 0 (T m + p + α) ≥ 1 we conclude. If h 0 (T m + p + α) = 0, then h 0 (T m + α + p + q) = 1. Then there exists an effective divisor E such that E ∼ T m + p + q + α. Then 2E ∼ 2(T m + p + q). As before, it follows h 0 (2(T m + p + q)) ≥ 2.

Surjecitivty for general curves
Let M g,d be the Deligne-Mumford stack of smooth curves of genus g with d unordered points. Let R g be the stack of Prym curves with genus g. We consider the stack of Prym curves of genus g with d unordered points: (5.1) R g,d := R g × Mg M g,d .
and so we conclude that coker(Φ ω C −T d ,ω C −T d +α ) = 0 for the general element. The proof for Φ C,ω C ,ω C −T d +α is analogous.
Analogously, an easy calculation shows that in order to have the surjectivity of 0.3, we need d ≤ g−3, if g = 4 or g = 5, and d ≤ g − 5 2 − √ 8g − 7 + cork(Φ 0 ′ ) if g ≥ 6. We plan to return to the problem in the next future and improve the conditions on g and d to have the surjectivity of the Gaussian maps.