We determine the number of minimal degree covers of odd ramification for a general curve.Moschetti is member of GNSAGA (INDAM) and is partially supported by MIUR: Dipartimenti di Eccellenza Program (2018-2022)-Dept. of Math. Univ. of Pavia; Naranjo was partially supported by the Proyecto de Investigación MTM2015-65361-P; Pirola is member of GNSAGA (INDAM) and is partially supported by PRIN Project Moduli spaces and Lie theory (2017) and by MIUR: Dipartimenti di Eccellenza Program (2018-2022) -Dept. of Math. Univ. of Pavia. Farkas was supported by the DFG Grant Syzygien und Moduli. PRELIMINARIESWe collect a few things that will be used throughout the paper.2.1. Monodromy of coverings and Hurwitz spaces of odd covers. Let f : C → 1 be a finite cover of degree d and denote by B := {P 1 , . . ., P n } its branch locus. For a point Q ∈ 1 \ B , letbe its monodromy representation. We denote by M f := Im(ρ f ) the monodromy group of f . The local monodromy of f around a branch point P i ∈ B is given by τ, where γ i is a simple loop around P i based at Q . The cover f is said to be alternating if M f ⊆ A d . We shall often consider alternating covers f : C → 1 , such that each local monodromy τ i is given by an odd cycle. We refer to such an f as being an odd cover.We denote by odd g the Hurwitz space parametrizing odd covers f : C → 1 of degree 2g + 1 branched at 3g points. We require that the local monodromy around each branch point of f be given by a 3-cycle. Such a cover is endowed with a theta characteristics ϑ := C (D ) ⊗ f * ( 1 (−1)),

We classify coherent modules on k [x , y ] of length at most 4 and supported at the origin. We compare our calculation with the motivic class of the moduli stack parametrizing such modules, extracted from the Feit-Fine formula. We observe that the natural torus action on this stack has finitely many fixed points, corresponding to connected skew Ferrers diagrams.

We describe an Azumaya algebra on the resolution of singularities of the double cover of a plane ramified along a nodal sextic associated to a non generic cubic fourfold containing a plane. We show that the derived category of such a resolution, twisted by the Azumaya algebra, is equivalent to the Kuznetsov component in the semiorthogonal decomposition of the derived category of the cubic fourfold.

Abstract. We show that every exact fully faithful functor from the category of perfect complexes on the spectrum of dual numbers to the bounded derived category of a noetherian separated scheme is of Fourier-Mukai type. The kernel turns out to be an object of the bounded derived category of coherent complexes on the product of the two schemes. We also study the space of stability conditions on the derived category of the spectrum of dual numbers.

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