The Divisors of Prym Semicanonical Pencils

Abstract

In the moduli space ${{\mathcal {R}}}_g$ of double étale covers of curves of a fixed genus $g$, the locus of covers of curves with a semicanonical pencil decomposes as the union of two divisors—${{\mathcal {T}}}^e_g$ and ${{\mathcal {T}}}^o_g$. Adapting arguments of Teixidor for the divisor of curves having a semicanonical pencil, we prove that both divisors are irreducible and compute their cohomology classes in the Deligne–Mumford compactification ${\overline {{\mathcal {R}}}}_g$.

Geometry of Prym semicanonical pencils and an application to cubic threefolds

“…𝑔 and  𝑜 𝑔 in the rational Picard group Pic( 𝑔 ) ℚ have recently been computed in [28]. The reader is referred to [19,Section 1] for the definition of the classes 𝜆, 𝛿 ′ 0 , 𝛿 ′′ 0 , 𝛿 ram 0 , 𝛿 𝑖 , 𝛿 𝑔−𝑖 , 𝛿 𝑖∶𝑔−𝑖 (1 ≤ 𝑖 ≤ [𝑔∕2]) generating Pic( 𝑔 ) ℚ .…”

“…We do not specify the coefficients of 𝛿 𝑖 , 𝛿 𝑔−𝑖 , 𝛿 𝑖∶𝑔−𝑖 since they are not useful for us. Theorem 2.1 [28]. Let [ 𝑒 𝑔 ], [ 𝑜 𝑔 ] ∈ Pic( 𝑔 ) ℚ denote the cohomology classes of  𝑒 𝑔 ,  𝑜 𝑔 in the Deligne-Mumford compactification  𝑔 .…”

“…The classes of $$\mathcal {T}^e_g$$ and $$\mathcal {T}^o_g$$ in the rational Picard group $$\operatorname{Pic}(\overline{\mathcal {R}}_g)_\mathbb {Q}$$ have recently been computed in [28]. The reader is referred to [19, Section 1] for the definition of the classes $$\lambda , \delta _0^{\prime }, \delta _0^{\prime \prime }, \delta _0^{\text{ram}}, \delta _i, \delta _{g-i},\delta _{i:g-i}$$ ($$1\le i\le [g/2]$$) generating $$\operatorname{Pic}(\overline{\mathcal {R}}_g)_\mathbb {Q}$$.…”

“…In the case of smooth curves of genus 3, a semicanonical pencil is the same as a $$g^1_2$$, so the divisor $$\mathcal {T}_3\subset \mathcal {M}_3$$ equals the hyperelliptic locus $$\mathcal {H}_3$$. At the level of Prym curves, the irreducible divisors $$\mathcal {T}^e_3$$ and $$\mathcal {T}^o_3$$ admit an easy description in terms of the number of Weierstrass points needed to express the 2‐torsion line bundle (see [28, Example 2.1]).…”

“…Recently, Maestro and the third author have proved in [28] that  𝑒 𝑔 and  𝑜 𝑔 are irreducible for 𝑔 ≠ 4, and have computed their classes in the rational Picard group of  𝑔 . The irreducibility for 𝑔 = 4 will be obtained as a consequence of our results.…”