Homaloidal nets and ideals of fat points. I (Q2804225)

The authors study plane Cremona maps with proper base points and the base ideal generated by the linear system of forms defining the map. Before we formulate some results for Cremona maps (including the main result for a plane Cremona map of degree \(d \geq 4\)) we need to recall some definitions. Assume that \(k\) is an infinite field. A linear system of plane curves of degree \(d\) is a \(k\)-vector subspace \(L_{d}\) of the vector space of forms of degree \(d\) in the standard graded polynomial ring \(R := k[x,y,z]\). A linear system \(L_{d}\) defines a rational map \(\mathcal{L}_{d} : \mathbb{P}^{2} \rightarrow \mathbb{P}^{r}\), where \(r+1\) is the vector space dimension of \(L_{d}\). If a map is birational, then it is called a Cremona map. Write \(I = (I : F)F\), where \(F \in R\) is, up to non-zero scalars, a uniquely defined form of degree \( \leq d\) such that \(I : F\) has codimension greater or equal to two. Then \(F\) is the fixed part of the system. We say that the linear system has no fixed part meaning that \(\text{deg} \,F = 0\). Here in the paper the authors assume that \(I : F\) has codimension equal to two.NEWLINENEWLINEGiven a variety \(X\), a smooth point \(P\) and a divisor \(D\) then the multiplicity \(e_{P}(D)\) is defined as NEWLINE\[NEWLINEe_{P}(D) = \text{max} \{s \geq 0 : f \in \mathfrak{m}_{P}^{s}\},NEWLINE\]NEWLINE where \(f\) is a local equation of \(D\) and \(\mathfrak{m}_{P}\) is the maximal ideal of the local ring of \(X\) at \(P\).NEWLINENEWLINEThe virtual multiplicity of \(L_{d}\) at one of its proper base points \(P\) is defined as NEWLINE\[NEWLINE\mu_{P} = \mu_{P}(L_{d}) : = \text{min}\{e_{P}(f) : f \in L_{d}\}.NEWLINE\]NEWLINE An important property of a plane Cremona maps is that its virtual multiplicities satisfy the classical equations of condition NEWLINE\[NEWLINE\sum_{P} \mu_{P} = 3d-3, \,\,\,\, \sum_{P} \mu_{P}^{2} = d^{2} - 1,NEWLINE\]NEWLINE where \(P\) runs over the set of base points of the corresponding \(L_{d}\) with respect to multiplicities \(\mu_{P}\). An abstract configuration \((d, \mu_{1},\dots, \mu_{r})\) satisfying the equations of condition is called a homaloidal type. A homaloidal type is called proper if there exists a plane Cremona map with this type.NEWLINENEWLINEA de Jonquiéres map is a plane Cremona map \(\mathfrak{F}\) of degree \(d\geq 2\) having the homaloidal type \((d; d-1, 1^{2d-2})\).NEWLINENEWLINENow we are given a set \(\mathcal{P} = \{P_{1}, \dots, P_{n} \} \subset \mathbb{P}^{2}\) of points with multiplicities \((\mu_{1}, \dots, \mu_{n})\). The \(\mu\)-fat ideal of \(\mathcal{P}\) is given by NEWLINE\[NEWLINEI(\mathcal{P}; \mu) = \bigcap_{i=1}^{n} I(P_{i})^{\mu_{i}},NEWLINE\]NEWLINE where \(I(P_{i})\) is the homogeneous prime ideal of the point \(P_{i}\). Recall that a net of degree \(d\) is a linear system \(L_{d}\) spanned by three independent forms in \(R_{d}\) without a proper common factor. We say that the net defined by \(L_{d}\) is complete if \(L_{d} = J_{d}\), where \(J = I(\mathcal{P}; \mu)\). At last, the net \(L_{d}\) is homaloidal if it defines a Cremona map of \(\mathbb{P}^{2}\).NEWLINENEWLINENow we are ready to present the main result of the paper.NEWLINENEWLINETheorem. Let \(\mathfrak{F} : \mathbb{P}^{2} \dashrightarrow \mathbb{P}^{2}\) be a Cremona map with homaloidal type \((d; \mu)\), with \(d \geq 4\), whose base points \(\mathcal{P}\) are proper and let \(I\) denote its base ideal. Let \(J: = I(\mathcal{P};\mu)\) denote the associated ideal of fat points.NEWLINENEWLINEa) The minimal graded free resolution of \(J\) is of the form NEWLINE\[NEWLINE0 \rightarrow R(-(d+2))^{d-2} \oplus R(-(d+1))^{s} \rightarrow R(-(d+1))^{d-4+s} \oplus R(-d)^{3} \rightarrow J \rightarrow 0,NEWLINE\]NEWLINE where \(s\) is the number of independent linear syzygies of the ideal \(I\). Furthermore, \(0\leq s \leq 1\), while \(s=1\) if and only if \(\mathfrak{F}\) is a de Jonquiéres map.NEWLINENEWLINEb) The linear system \(J_{d+1}\) defines a birational mapping of \(\mathbb{P}^{2}\) onto the image in \(\mathbb{P}^{d+4+s}\).NEWLINENEWLINEMoreover, in Section 3 the authors present a partial classification of homaloidal types according to the highest virtual multiplicity. Some classical inequalities for Cremona map stem from the consideration of the three highest virtual multiplicities, but not much has been obtained by stressing the role of the behavior of the map in the neighborhood of a single point with highest virtual multiplicity \(\mu_{1} = \mu\), so the authors study this case in detail. For instance, they show that for \(\mu=d-2\) any Cremona map on general points of such a homaloidal type is saturated (Proposition 3.1). In the last section the authors study the following question.NEWLINENEWLINEQuestion. Let \((d, \mu_{1}, \dots)\) denote a proper homaloidal type and let \(F\) be a Cremona map of this type and general base points. If the base ideal of \(F\) is not saturated, then \(\mu_{1} \leq \lfloor d/2 \rfloor\).NEWLINENEWLINEAs a result the authors show that the base ideal \(I\) of a Cremona map \(\mathfrak{F} : \mathbb{P}^{2} \dashrightarrow \mathbb{P}^{2}\) of degree \(d\), whose virtual multiplicities are all even, is not saturated under the hypothesis that there are three virtual multiplicities equal to \(\lfloor d/2 \rfloor\).