Surfaces parameterizing Waring presentations of smooth plane cubics
We start with a cubic form \(\varphi\) on a three-dimensional vector space \(V\) which cuts out a smooth curve \(C\) in \(P(V)\). Assume that the \(j\)-invariant of \(C\) is nonzero, that is, \(\varphi\) cannot be written as a sum of three cubes of linear forms. Then \(\varphi\) can be written as a sum of four cubes: (1) \(\varphi = l^3_1 + \cdots + l^3_4\). In this paper we shall study the set \(X_\varphi\) of all presentations (1). We show that \(X_\varphi\) is an irreducible affine algebraic surface of general type. Moreover, we prove that its minimal compactification \(\overline X_\varphi\) has positive index. The rest of the paper is structured as follows. Section 2 contains some preliminary facts about plane cubics. In section 3 we prove an irreducibility criterion for abelian coverings of smooth irreducible varieties. In section 4 we give an explicit description of the surface \(X_\varphi\) by relating it to the Hessian curve \(H\) of \(\varphi\). The surface \(X_\varphi\) is then realized as an abelian cover of an open subset of \(H \times H\). In section 5 we use this construction and the criterion of section 3 to prove the irreducibility of \(X_\varphi\). We also prove that the generic fiber of the map \(X_\varphi \to H\) is irreducible. -- Next we turn to numerical invariants. In section 6 and 7 we present an explicit construction of the minimal compactification \(\overline X_\varphi\) of \(X_\varphi\) and show that the genus of the generic fiber of \(X_\varphi \to H\) is 325. In section 8 we show that the Chern numbers of \(\overline X_\varphi\) are given by \(c^2_1 = 7,452\) and \(c_2 = 2,916\) and thus \(\overline X_\varphi\) has positive index.
scientific article; zbMATH DE number 488358