We compute the dimension of the image of the map $$\pi _Z :Z \rightarrow {\mathcal {M}}_g$$ π Z : Z → M g forgetting the markings, where Z is a connected component of the stratum $${\mathcal {H}}^k_g(\mu )$$ H g k ( μ ) of k-differentials with an assigned partition $$\mu $$ μ , for the cases when $$k=1$$ k = 1 with meromorphic partition and $$k=2$$ k = 2 when the quadratic differentials have at worst simple poles.

We prove that for a generic element in a nonhyperelliptic component of an abelian stratum Hg(μ) in genus g, the underlying curve has maximal gonality. We extend this result to the case of quadratic strata when the partition μ has positive entries. As a consequence we deduce that all nonhyperelliptic components of H9(μ) are uniruled when μ is a positive partition of 16 and all nonhyperelliptic components of H 2 g (μ) are uniruled when μ is a positive partition of 4g − 4 and either 3 g 5 or g = 6 and l(μ) 4.

For genus $g = \frac {r(r+1)}{2}+1$, we prove that via the forgetful map, the universal Prym–Brill–Noether locus $\mathcal {V}^r_g$ has a unique irreducible component dominating the moduli space $\mathcal {R}_g$ of Prym curves.

For even genus $$g=2i\ge 4$$ g = 2 i ≥ 4 and the length $$g-1$$ g - 1 partition $$\mu = (4,2,\ldots ,2,-2,\ldots ,-2)$$ μ = ( 4 , 2 , … , 2 , - 2 , … , - 2 ) of 0, we compute the first coefficients of the class of $$\overline{D}(\mu )$$ D ¯ ( μ ) in $$\mathrm {Pic}_{\mathbb {Q}}(\overline{{\mathcal {R}}}_g)$$ Pic Q ( R ¯ g ) , where $$D(\mu )$$ D ( μ ) is the divisor consisting of pairs $$[C,\eta ]\in {\mathcal {R}}_g$$ [ C , η ] ∈ R g with $$\eta \cong {\mathcal {O}}_C(2x_1+x_2+\cdots + x_{i-1}-x_i-\cdots -x_{2i-1})$$ η ≅ O C ( 2 x 1 + x 2 + ⋯ + x i - 1 - x i - ⋯ - x 2 i - 1 ) for some points $$x_1,\ldots , x_{2i-1}$$ x 1 , … , x 2 i - 1 on C. We further provide several enumerative results that will be used for this computation.

For genus g = 2i ≥ 4 and the length g − 1 partition µ = (4, 2, . . . , 2, −2, . . . , −2) of 0, we compute the first coefficients of the class offor some points x 1 , . . . , x 2i−1 on C. We further provide several enumerative results that will be used for this computation.