On the finite generation of the effective monoid of rational surfaces

Abstract

a b s t r a c tWe give a numerical criterion for ensuring the finite generation of the effective monoid of the surfaces obtained by a blowing-up of the projective plane at the supports of zero dimensional subschemes assuming that these are contained in a degenerate cubic. Furthermore, this criterion also ensures the regularity of any numerically effective divisor on these surfaces. Thus the dimension of any complete linear system is computed. On the other hand, in particular and among these surfaces, we obtain ringed rational surfaces with very large Picard numbers and with only finitely many integral curves of strictly negative self-intersection. These negative integral curves except two (−1)-curves are all contained in the support of an anticanonical divisor. Thus almost all the geometry of such surfaces is concentrated in the anticanonical class.

On the effective, nef, and semi-ample monoids of blowups of Hirzebruch surfaces at collinear points

“…Such results were published after in [37]. Nowadays, such problem is still open even in the case of surfaces, some contributions in this direction are [9], [21], [4], [30], [15], [31], [17], [3], [33], [10], [11], [12], [32], [18], [13], [14], [6], [19] and [34].…”

Rational Surfaces with Anticanonical Divisor not Reduced

“…The case where r is less than ten is studied in [25], [23], [15], [17], [18] and [19]. Some cases where r is greater than nine have been studied in [10], [11], [14], [2], [5], [6], [7], [8], [9] and [21].…”