ALGEBRAIC HYPERBOLICITY OF RAMIFIED COVERS OF G(m)(2) (AND INTEGRAL POINTS ON AFFINE SUBSETS OF P-2)

作者：Corvaja Pietro^*; Zannier Umberto

来源：Journal of Differential Geometry, 2013, 93(3): 355-377.

Let X be a smooth affine surface, X -%26gt; G(m)(2) be a finite morphism. We study the affine curves on X, with bounded genus and number of points at infinity, obtaining bounds for their degree in terms of Euler characteristic. %26lt;br%26gt;A typical example where these bounds hold is represented by the complement of a three-component curve in the projective plane, of total degree at least 4. %26lt;br%26gt;The corresponding results may be interpreted as bounding the height of integral points on X over a function field. In the language of Diophantine Equations, our results may be rephrased in terms of bounding the height of the solutions of f (u, v, y) = 0, with u, v, y over a function field, u, v S-units. %26lt;br%26gt;It turns out that all of this contain some cases of a strong version of a conjecture of Vojta over function fields in the split case. Moreover, our method would apply also to the nonsplit case. %26lt;br%26gt;We remark that special cases of our results in the holomorphic context were studied by M. Green already in the seventies, and recently in greater generality by Noguchi, Winkelmann, and Yamanoi [12]; however, the algebraic context was left open and seems not to fall in the existing techniques.

• 出版日期2013-3