On blowing-up of polarized surfaces

Rational conic fibrations of sectional genus two

“…Note that ν > 7 for all admissible pairs (x, y) with x ≥ 3, hence requirement ( 12) is meaningful only for the restricted subset of admissible (x, y) with x ≤ 2. Moreover, ( 12) is obvious when x = 0 because the seven points lie on distinct fibers; similarly, (12) is obvious also when y = 0, since in this case C = s or 2s according to whether x = 1 or 2, and we know that at most one of the p i 's can lie on s. This reduces the analysis to 11 − 5 = 6 admissible pairs and a close check can be done. In particular, recalling that µ = 7, in addition to the conditions in j) of Theorem 10, we see that (12) includes some further restrictions, like that the seven point cannot lie on a curve in |s + 3f | or in |2s + 3f |.…”

“…Moreover, ( 12) is obvious when x = 0 because the seven points lie on distinct fibers; similarly, (12) is obvious also when y = 0, since in this case C = s or 2s according to whether x = 1 or 2, and we know that at most one of the p i 's can lie on s. This reduces the analysis to 11 − 5 = 6 admissible pairs and a close check can be done. In particular, recalling that µ = 7, in addition to the conditions in j) of Theorem 10, we see that (12) includes some further restrictions, like that the seven point cannot lie on a curve in |s + 3f | or in |2s + 3f |. Looking at the plane model and using the usual notation, we see that these curves are the proper transforms on F 1 of a cubic passing through q 0 with multiplicity 2 or 1 respectively.…”

“…. , p µ are general, then L is ample, according to Yokoyama [12,Theorem,1.8]. Then (X, L) is a polarized rational conic fibration of sectional genus 2.…”

“…. , p µ are general enough, according to Yokoyama [12]. So, as a first thing we explore the precise conditions that the points p 1 , .…”

Generalized polarized manifolds with low second class

“…The aim of this paper is twice. First of all, starting from Table 1, which combines the list in [30,Theorem 4.3] with results of Fujita [6] and Yokoyama [35], summarizing what is known for polarized surfaces with m À d a 8, we prove new results concerning the case 2g þ 1 a m À d a 2g þ 2, where g is the sectional genus (Theorem 1) and the case of polarized nonruled surfaces with m À 2d < 2g (Proposition 1). Both will play a relevant role in the sequel.…”