Blown-Up Hirzebruch Surfaces and Special Divisor Classes

We work on special divisor classes on blow-ups F p , r of Hirzebruch surfaces over the field of complex numbers, and extend fundamental properties of special divisor classes on del Pezzo surfaces parallel to analogous ones on surfaces F p , r . We also consider special divisor classes on surfaces F p , r with respect to monoidal transformations and explain the tie-ups among them contrast to the special divisor classes on del Pezzo surfaces. In particular, the fundamental properties of quartic rational divisor classes on surfaces F p , r are studied, and we obtain interwinded relationships among rulings, exceptional systems and quartic rational divisor classes along with monoidal transformations. We also obtain the effectiveness for the rational divisor classes on F p , r with positivity condition.

A TORELLI THEOREM FOR SPECIAL DIVISOR VARIETIES X ASSOCIATED TO DOUBLY COVERED CURVES ${\tilde C}/C$

Let Ci, i=1,2, be two smooth non-hyperelliptic curves over the complex numbers of genus g≥ 3, and [Formula: see text] connected étale double covers, such that the theta divisors Ξi of the associated Prym varieties (pi, Ξi) are non singular in codimension ≤ 3. If [Formula: see text] are the norm maps, then Ξi is isomorphic to {[Formula: see text] and h0 (L) is even and positive}. Then the Abel maps define generic ℙ1 bundles Xi→Ξi, where Xi is the special divisor variety [Formula: see text] and [Formula: see text] even}. We prove, under the hypotheses above, that biregular isomorphism of the special divisor varieties X1≅ X2 implies isomorphism of the double covers [Formula: see text].