Embeddings of general blowing-ups at points.

*(English)*Zbl 0833.14026In their paper in Math. Z. 211, No. 3, 479-483 (1992; Zbl 0759.14004), J. D’Almeida and A. Hirschowitz prove the following theorem: Let \(t \in \mathbb{Z}_{\geq 1}\) and let \(P_1; \ldots; P_k\) be \(k \leq ((t^2 + 3t)/2) - 5\) general points on the projective plane \(\mathbb{P}^2 \). Let \(\pi : M \to \mathbb{P}^2\) be the blowing-up of \(\mathbb{P}^2\) at those points and let \(E\) be the exceptional divisor. Then the invertible sheaf \({\mathcal L} = \pi^* ({\mathcal O}_{\mathbb{P}^2} (t)) \otimes {\mathcal O}_M (-E)\) is very ample.

In this paper we generalize this result in two directions. First of all, we consider the projective space \(\mathbb{P}^n\) for any \(n \geq 2\) and we prove the following. Let \(t \in \mathbb{Z}_{\geq 1}\) and let \(P_1; \ldots; P_k\) be \(k \leq {n + t \choose t} - (n - 1)(n+1) - 4\) general points on \(\mathbb{P}^n\). Let \(\pi : M \to \mathbb{P}^n\) be the blowing-up of \(\mathbb{P}^n\) at those points and let \(E\) be the exceptional divisor. Then the invertible sheaf \({\mathcal L} = \pi^* ({\mathcal O}_{\mathbb{P}^n} (t)) \otimes {\mathcal O}_M (-E)\) is very ample.

For \(n = 2\) this statement is one possible value weaker than the above mentioned theorem. However, our proof is completely different and in the case \(n = 2\) it is much more elementary than the proof of d’Almeida and Hirschowitz. Moreover in case \(n = 2\) we describe how to prove the theorem for \(k = ((t^2 + 3t)/2) - 5\) using our methods. Also, we obtain the theorem for special types of non general configurations of the points and for those configurations our theorem is sharp.

In another direction we generalize the theorem of d’Almeida and Hirschowitz for arbitrary surfaces as follows. Let \(X\) be a smooth projective surface and let \({\mathcal M}\) be a very ample invertible sheaf on \(X\). Let \(t \in \mathbb{Z}_{ \geq 1}\) with \(t \geq 7\) and let \(P_1; \ldots; P_k\) be \(k \leq \dim (\Gamma (X; {\mathcal M}^{\otimes t})) - 7\) general points on \(X\). Let \(\pi : M \to X\) be the blowing-up of \(X\) at \(P_1; \ldots; P_k\) and let \(E\) be the exceptional divisor on \(M\) blowing- down by \(\pi\). The invertible sheaf \(\pi^* ({\mathcal M}^{\otimes t}) \otimes {\mathcal O}_M (-E)\) is very ample. – In particular taking \(k = \dim (\Gamma (X; {\mathcal M}^{\otimes t})) - 7\) we find a linearly normal embedding (i.e. by means of a complete linear system) of a smooth surface \(M\) birationally equivalent to \(X\) in \(\mathbb{P}^6\). The optimal result in this direction would be that each surface \(X\) has a smooth linearly normal birational model in \(\mathbb{P}^5\). More general, one can ask the question whether each smooth projective variety \(X\) of dimension \(n\) has a smooth linearly normal birational model in \(\mathbb{P}^{2n + 1}\).

An important ingredient in most of the proofs is a generalization of the well-known trisecant lemma for curves. In particular, we use and prove the following result. Let \(h\) be an \(r\)-dimensional very ample linear system on a smooth complete curve \(C\). Assume the dimension of the space parametrizing \(f\)-secant \((f - 2)\)-spaces has dimension at least \(f-2\) for some \(5 \leq f \leq r - 1\), then the dimension of the space parametrizing 5-secant 3-spaces has dimension at least 3.

In this paper we generalize this result in two directions. First of all, we consider the projective space \(\mathbb{P}^n\) for any \(n \geq 2\) and we prove the following. Let \(t \in \mathbb{Z}_{\geq 1}\) and let \(P_1; \ldots; P_k\) be \(k \leq {n + t \choose t} - (n - 1)(n+1) - 4\) general points on \(\mathbb{P}^n\). Let \(\pi : M \to \mathbb{P}^n\) be the blowing-up of \(\mathbb{P}^n\) at those points and let \(E\) be the exceptional divisor. Then the invertible sheaf \({\mathcal L} = \pi^* ({\mathcal O}_{\mathbb{P}^n} (t)) \otimes {\mathcal O}_M (-E)\) is very ample.

For \(n = 2\) this statement is one possible value weaker than the above mentioned theorem. However, our proof is completely different and in the case \(n = 2\) it is much more elementary than the proof of d’Almeida and Hirschowitz. Moreover in case \(n = 2\) we describe how to prove the theorem for \(k = ((t^2 + 3t)/2) - 5\) using our methods. Also, we obtain the theorem for special types of non general configurations of the points and for those configurations our theorem is sharp.

In another direction we generalize the theorem of d’Almeida and Hirschowitz for arbitrary surfaces as follows. Let \(X\) be a smooth projective surface and let \({\mathcal M}\) be a very ample invertible sheaf on \(X\). Let \(t \in \mathbb{Z}_{ \geq 1}\) with \(t \geq 7\) and let \(P_1; \ldots; P_k\) be \(k \leq \dim (\Gamma (X; {\mathcal M}^{\otimes t})) - 7\) general points on \(X\). Let \(\pi : M \to X\) be the blowing-up of \(X\) at \(P_1; \ldots; P_k\) and let \(E\) be the exceptional divisor on \(M\) blowing- down by \(\pi\). The invertible sheaf \(\pi^* ({\mathcal M}^{\otimes t}) \otimes {\mathcal O}_M (-E)\) is very ample. – In particular taking \(k = \dim (\Gamma (X; {\mathcal M}^{\otimes t})) - 7\) we find a linearly normal embedding (i.e. by means of a complete linear system) of a smooth surface \(M\) birationally equivalent to \(X\) in \(\mathbb{P}^6\). The optimal result in this direction would be that each surface \(X\) has a smooth linearly normal birational model in \(\mathbb{P}^5\). More general, one can ask the question whether each smooth projective variety \(X\) of dimension \(n\) has a smooth linearly normal birational model in \(\mathbb{P}^{2n + 1}\).

An important ingredient in most of the proofs is a generalization of the well-known trisecant lemma for curves. In particular, we use and prove the following result. Let \(h\) be an \(r\)-dimensional very ample linear system on a smooth complete curve \(C\). Assume the dimension of the space parametrizing \(f\)-secant \((f - 2)\)-spaces has dimension at least \(f-2\) for some \(5 \leq f \leq r - 1\), then the dimension of the space parametrizing 5-secant 3-spaces has dimension at least 3.

Reviewer: M.Coppens (Geel)

##### MSC:

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

14E22 | Ramification problems in algebraic geometry |

14N05 | Projective techniques in algebraic geometry |

14J17 | Singularities of surfaces or higher-dimensional varieties |