A remark on the intersection of plane curves

Abstract

Let D be a very general curve of degree d = 2ℓ − ε in P 2 , with ε ∈ {0, 1}. Let Γ ⊂ P 2 be an integral curve of geometric genus g and degree m, Γ = D, and let ν : C → Γ be the normalization. Let δ be the degree of the reduction modulo 2 of the divisor ν * (D) of C (see § 2.1). In this paper we prove the inequality 4g + δ m(d − 8 + 2ε) + 5. We compare this with similar inequalities due to Geng Xu ([88, 89]) and Xi Chen ([17, 18]). Besides, we provide a brief account on genera of subvarieties in projective hypersurfaces.

Lines, conics, and all that

“…Conjecture. (C. Ciliberto, F. Flamini, and M. Zaidenberg [95]) There exists a strictly growing function ϕ : N → N such that the number of curves of geometric genus g ϕ(d) in any smooth surface S of degree d 5 in P 3 is finite and bounded by a function of d.…”