The Effect of Points Fattening on del Pezzo Surfaces

Bulletin of the Malaysian Mathematical Sciences Society ◽

10.1007/s40840-020-01018-2 ◽

2020 ◽

Author(s):

Magdalena Lampa-Baczyńska

Keyword(s):

Complex Numbers ◽

Del Pezzo Surfaces ◽

Minimal Growth ◽

Complete Answer ◽

Initial Sequence ◽

Blowing Up ◽

Del Pezzo

Abstract In this paper, we study the fattening effect of points over the complex numbers for del Pezzo surfaces $$\mathbb {S}_r$$ S r arising by blowing-up of $$\mathbb {P}^2$$ P 2 at r general points, with $$ r \in \{1, \dots , 8 \}$$ r ∈ { 1 , ⋯ , 8 } . Basic questions when studying the problem of points fattening on an arbitrary variety are what is the minimal growth of the initial sequence and how are the sets on which this minimal growth happens characterized geometrically. We provide a complete answer for del Pezzo surfaces.

Download Full-text

Blown-Up Hirzebruch Surfaces and Special Divisor Classes

Mathematics ◽

10.3390/math8060867 ◽

2020 ◽

Vol 8 (6) ◽

pp. 867

Author(s):

Jae-Hyouk Lee ◽

YongJoo Shin

Keyword(s):

Complex Numbers ◽

Del Pezzo Surfaces ◽

Positivity Condition ◽

Fundamental Properties ◽

Special Divisor ◽

Hirzebruch Surfaces ◽

Del Pezzo

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.

Download Full-text

BOUNDING THE VOLUMES OF SINGULAR WEAK LOG DEL PEZZO SURFACES

International Journal of Mathematics ◽

10.1142/s0129167x13501103 ◽

2013 ◽

Vol 24 (13) ◽

pp. 1350110 ◽

Cited By ~ 2

Author(s):

CHEN JIANG

Keyword(s):

Upper Bound ◽

General Position ◽

Del Pezzo Surfaces ◽

Minimal Resolution ◽

Picard Number ◽

Hirzebruch Surface ◽

Del Pezzo Surface ◽

Blowing Up ◽

Del Pezzo ◽

Log Del Pezzo Surfaces

We give an optimal upper bound for the anti-canonical volume of an ϵ-lc weak log del Pezzo surface. Moreover, we consider the relation between the bound of the volume and the Picard number of the minimal resolution of the surface. Furthermore, we consider blowing up several points on a Hirzebruch surface in general position and give some examples of smooth weak log del Pezzo surfaces.

Download Full-text

Del Pezzo surfaces over finite fields and their Frobenius traces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004118000166 ◽

2018 ◽

Vol 167 (01) ◽

pp. 35-60 ◽

Cited By ~ 2

Author(s):

BARINDER BANWAIT ◽

FRANCESC FITÉ ◽

DANIEL LOUGHRAN

Keyword(s):

Finite Field ◽

Finite Fields ◽

Del Pezzo Surfaces ◽

Cubic Surface ◽

Inverse Galois Problem ◽

Complete Answer ◽

Cubic Surfaces ◽

Analogous Question ◽

Special Cases ◽

Del Pezzo

AbstractLet S be a smooth cubic surface over a finite field $\mathbb{F}$q. It is known that #S($\mathbb{F}$q) = 1 + aq + q2 for some a ∈ {−2, −1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases treated by Swinnerton–Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton–Dyer's tables on cubic surfaces over finite fields.

Download Full-text