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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).

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[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.

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