Recovering Plane Curves from Their Bitangents

1.1. It is well known that a general plane curve of degree d has 12d(d − 2)(d − 9) distinct bitangent lines. The first (and most) interesting case is that of a smooth plane quartic X , whose configuration of 28 bitangents we shall denote by θ(X), to highlight the correspondence with the odd theta-characteristics of X . The properties of θ(X) have been extensively studied by the classical geometers since the time of Riemann (Aronhold, Cayley, Hesse, Plücker, Schottky, Steiner, Weber just to quote a few). Such an exceptional interest is not only due to its rich geometry, but also and especially to the connection of this configuration with the classical theory of theta functions. Some of the features of θ(X) discovered classically have to do with the possibility of recovering the curve X from various data related to θ(X). The most celebrated of these results is due to Aronhold: he discovered for every nonsingular quartic X the existence of 288 7-tuples of bitangents, called Aronhold systems, characterized by the condition that for any three lines of such a system the six points of contact are not on a conic; he showed that the full configuration θ(X) and the curve X can be reconstructed from any one of these Aronhold systems. For an account of his construction we refer to [E-Ch], p. 319, or [K-W], p. 783. Notice that the definition of Aronhold system requires knowing not only θ(X), but also the contact points between X and its bitangents. And since there is no known way to recognize the Aronhold systems only knowing θ(X), the result of Aronhold leaves still unanswered the question below, to which this paper is devoted: