On the variety of the inflection points of plane cubic curves

Abstract

In the paper, we investigate properties of the nine-dimensional variety of the inflection points of the plane cubic curves. The description of local monodromy groups of the set of inflection points near singular cubic curves is given. Also, it is given a detailed description of the normalizations of the surfaces of the inflection points of plane cubic curves belonging to general two-dimensional linear systems of cubic curves, The vanishing of the irregularity a smooth manifold birationally isomorphic to the variety of the inflection points of the plane cubic curves is proved.

Arithmetic inflection of superelliptic curves

“…12 Kulikov [14] has solved the analogue of this problem for two-dimensional families of plane curves subject to a genericity hypothesis. Here New(∆ 5,3 ) = 5New(∆ * ), and as New(∆ 5,3 ) contains 131 interior lattice points, it follows that the arithmetic genus of the closure of ∆ 5,3 in Tor(∆ * ) is 131.…”

Rational differential forms on the variety of flexes of plane cubics

“…The closed subset X of P 2 × P 9 , defined by the system of equations F = H = 0 was explored in the papers [4], [6], [7]. It is irreducible, nine-dimensional and has singularities (as a matter of fact, X even is non-normal) [7]. Let P 2 p 2 ← − X p 9 − → P 9 be the natural projections.…”

“…Let Y be a smooth irreducible projective algebraic variety birationally isomorphic to the algebraic variety X. The main result (Theorem 4) of the paper [7] is the claim that the irregularity of the variety Y vanishes:…”