# On rational surfaces, II

@inproceedings{Nagata1960OnRS,
title={On rational surfaces, II},
author={Masayoshi Nagata},
year={1960}
}
In § 1 of the present paper, we introduce the notion of a virtual linear system on a non-singular projective surface and we clarify the theories o f infinitely near points, of divisors and o f linear system with preassigned base conditions. We introduce in § 2 the notions of a numerical types and of non-special points with respect to Cremona transformations. They play important roles in § 3 in order to prove characterizations and existence theorems o f exceptional curves of the first kind and…
244 Citations

### A pr 2 00 1 EXISTENCE OF CURVES WITH PRESCRIBED TOPOLOGICAL SINGULARITIES

• Mathematics
• 2001
Throughout this paper we study the existence of irreducible curves C on smooth projective surfaces Σ with singular points of prescribed topological types S 1 ,. .. , S r. There are necessary

### Complete linear systems on rational surfaces

We determine the dimension, fixed components and base points of complete linear systems on blowings-up of P2 having irreducible anticanonical divisor. Consider a set of pointspx,...,p„ of the

### RATIONAL SURFACES OVER PERFECT FIELDS. II

Letk be a perfect field of arbitrary characteristic. The main object of this paper is to establish some new objects associated with algebraic surfaces F defined overk which are invariants for

### Linear systems on a special rational surface

We study the Hilbert series of a family of ideals J_\phi generated by powers of linear forms in k[x_1,...,x_n]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints

### On finiteness of curves with high canonical degree on a surface

• Mathematics
• 2014
The canonical degree of a curve C on a surface X is $$K_X\cdot C$$KX·C. Our main result, Theorem 1.1, is that on a surface of general type there are only finitely many curves with negative

### Plane Cremona maps, exceptional curves and roots

We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection -1 and -2, respectively) on a smooth complex projective surface S

### THE GEOMETRY OF RATIONAL SURFACES AND HILBERT FUNCTIONS OF POINTS IN THE PLANE

We study the structure of the set of numerically effective divisor classes on a rational surface and apply this to study hilbert functions of the homogeneous coordinate rings of 0-cycles on curves of

### 2 6 M ar 2 00 2 The Projective Theory of Ruled Surfaces

• Mathematics
• 2002
Introduction: Through this paper, a geometrically ruled surface, or simply a ruled surface, will be a P-bundle over a smooth curve X of genus g. It will be denoted by π : S = P(E0)−→X and we will

## References

SHOWING 1-8 OF 8 REFERENCES

### Franchetta, Sulle curve eccezionali di prima specie appartenenti ad una superficie algebrica

• Boll. Un. Mat. Ital.,
• 1940

### Recerche generali sopra i systemi lineari d i curve piani, Mem

• Accad. Sci. Torino,
• 1892

### Recerche generali sopra i systemi lineari d i curve piani

• Mem. Accad. Sci. Torino, ser

### Sulle curve eccezionali di prim a specie appartenenti ad una superficie algebrica

• Boll. Un. M at. Ital., ser

• 1958