Number of moduli of families of plane curves with nodes and cusps

In my Ph.D.-thesis I computed the number of moduli of certain families of plane curves with nodes and cusps. Let Σn k,d ⊂ P(H0(P2,OP2(n))) := PN, with N = n(n+3)2 , be the closure, in the Zariski’s topology, of the locally closed set of reduced and irreducible plane curves of degree n with k cusps and d nodes. We recall that, if k = 0, the varieties Vn,g = Σn0,d are called the Severi varieties of irreducible plane curves of degree n and geometric genus g = n−1 2 − d. Let Σ⊂ Σn k,d be an irreducible component of Σn k,d and let g = n−1 2 −d−k be the geometric genus of the plane curve corresponding to the general point of Σ. It is naturally defined a rational map ΠΣ : Σ Mg, sending the general point [Γ] ∈ Σ to the isomorphism class of the normalization of the plane curve Γ corresponding to the point [Γ]. We set number of moduli of Σ := dim(ΠΣ(Σ)). If k < 3n, then (1) dim(ΠΣ(Σ)) ≤ min(dim(Mg), dim(Mg) + ρ − k), where ρ := ρ(2, g, n) = 3n − 2g − 6 is the Brill-Neother number of the linear series of degree n and dimension 2 on a smooth curve of genus g. We say that Σ has the expected number of moduli if the equality holds in (1). By classical Brill-Neother theory when ρ is positive and by a well know result of Sernesi when ρ ≤ 0, we have that Σn0,d, (which is irreducible), has the expected number of moduli for every d ≤ n−1 2 . Working out the main ideas and techniques that Sernesi uses in [1], under the hypothesis k > 0, in my Ph.D.-thesis I find sufficient conditions in order that an irreducible component Σ ⊂ Σn k,d has the expected number of moduli. If Σ verifies these properties, then ρ ≤ 0. By using induction on the degree n and on the genus g of the general curve of the family, I prove that, if ρ ≤ 0 and k ≤ 6, then there exists at least one irreducible component of Σn k,d with expected number of moduli equal to 3g−3+ρ−k. By using this result and a result of Eisembud and Harris, from which it follows that, if ρ is positive enough and k ≤ 3 then dim(ΠΣ(Σ)) = 3g − 3, I prove that Σn1,d (which is irreducible) has the expected number of moduli for every d ≤ n−1 2 , i.e. for every ρ. I am extending this result to the case k ≤ 3. Finally, I consider the case of irreducible sextics with six cusps. It is classically know that Σ66,0 contains at least two irreducible components Σ1 and Σ2. The general point of Σ1 parametrizes a sextic with six cusps on a conic, whereas the general element of Σ2 corresponds to a sextic with six cusps not on a conic. I prove that Σ1 and Σ2 have expected number of moduli. I don’t still know example of irreducible complete families of plane curves with nodes and cusps having number of moduli smaller that the expected. Finally, in the first sections of my thesis, following essentially Zariski’s papers, I introduce classical techniques used to study and describe the geometry of a family of plane curves with assigned singularities. Then, I briefly resume the more modern results by Wahl on families of plane curves with nodes and cusps. I also give some applications of Horikawa deformation theory to the study of deformations of plane curves. Finally, I devoted a section of my thesis to the versal deformation family of plane curve singularity. In particular, by using the results of [3] and [2] and a simple argument of projective geometry, I proved that in the equigeneric locus of the ´etale versal deformation space B of an ordinary plane curve singularity there are only points corresponding to a plane curve with only ordinary multiple points. I mean that this result is known, but I haven’t found in literature a proof of this. References [1] E. Sernesi:On the existence of certain families of curves, Invent. math. vol. 75, (1984). [2] A. Morelli:Un’osservazione sulle singolarita’ delle trasformate birazionali di una curva algebrica, Rend. Acc. Sci. Napoli, serie 4 vol. 29 (1962), p.59-64. [3] A. Franchetta: Osservazioni sui punti doppi delle superfici algebriche, Rend. Acc. dei Lincei, gennaio 1946.