# zbMATH — the first resource for mathematics

Our thin knowledge of fat points. (English) Zbl 0743.14005
Curves Semin. at Queen’s. Vol. VI, Kingston/Canada 1989, Queen’s Pap. Pure Appl. Math. 83, Exposé B, 50 p. (1989).
[For the entire collection see Zbl 0695.00011.]
This paper is an expository paper that summarizes the contribution of many authors on the subject of “ fat points” in the projective plane. However it contains also some original results. A fat point is a $$0$$- dimensional scheme $$Z\subset\mathbb{P}^ 2(k)$$ given by intersection of powers of ideals of a finite set of points $$P_ 1,\ldots,P_ s$$, $$I=\bigcap^ s_{i=1}{\mathfrak p}_ i^{m_ i}$$. So $$\hbox{Supp}(Z)=\{P_ 1,\ldots,P_ s\}$$. If $$I=\oplus_{t\geq 0}I_ t$$, the linear space $$I_ t$$ gives the linear system of all curves of $$\hbox{degree} t$$ of at least multiplicity $$m_ i$$ at each $$P_ i$$. The nonnegative number $$S=h^ 1{\mathcal I}_ Z(t))$$ is called the superabundance of $$I_ t$$, the system $$I_ t$$ is called regular if $$S=0$$ and it is called superabundant if $$S>0$$. Many authors have studied fat points from the point of view of the dimension of linear systems of plane curves containing $$\hbox{Supp}(Z)$$. In the section 1 of the present paper are presented some results in connection with the following theorem of B. Segre: If the fat point $$Z$$ is contained in an integral plane curve $$C$$ of degree $$d$$ then it is the scheme theoretical complete intersection of $$C$$ with a curve $$C'$$ of degree $$d'$$ if and only if $$\deg(Z)=dd'$$ (Bézout condition) and $$H^ 1(\mathbb{P}^ 2,{\mathcal I}_ Z(d+d'-3))\neq 0$$ (Jacobi condition). The theorems (1.5) and (1.7) are generalisations of the theorem of Segre to more complicated multiplicities.
In the section 2, theorem (2.10) gives a bound of the degree of regular systems: if $$P_ 1,\ldots,P_ s$$ is a generic $$s$$-tuple of points in the plane and $$s\geq 5$$, $$m_ 1\geq m_ 2\geq\cdots\geq m_ s>0$$ then $\min\{t\mid I_ t \hbox{regular}\}\leq\max\{m_ 1+m_ 2- 1;\left[\sum^ s_{i=1}{m_ i \over 2}\right]; \sum^{m_ 1}_{i=1}d_ i\}$ where $$d_ i$$ are positive integers defined by $$m_ i$$’s. This is also a refinement of a result of Segre. Sections 3 and 4 deal with the blowing-up $$X_ s$$ of the projective plane in the points $$P_ 1,\ldots,P_ s$$. If $$E_ i$$ is the proper transformation of $$P_ i$$ and $$E_ 0$$ is the strict transformation of a line not containing any $$P_ i$$ the divisor $$D_ t=tE_ 0-m_ 1E_ 1-\cdots- m_ sE_ s$$ is defined. In the section 3, there is analyzed a conjecture of Hirschowitz on the non-speciality of $$D_ t$$ in order to obtain information about the regularity of $$I_ t$$. In the section 4 it is considered the problem to find values of $$t$$ for which the linear system $$| D_ t|$$ is very ample on $$X_ s$$. This one is a generalisation of a known problem in the case $$s\leq 6$$ (the case of Del Pezzo surfaces) or in the case $$m_ 1=\cdots=m_ s=1$$. In the theorem 4.3 it is proved the important original result: If $$s=3q+1$$ and $$m_ 1\geq q$$ then $$| D_{q+2}|$$ is very ample. So, $$X_ s$$ is a rational surface with hyperelliptic hyperplane sections.

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14M10 Complete intersections 14J25 Special surfaces 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
Zbl 0695.00011