Projective geometry of elliptic curves.

*(English)*Zbl 0602.14024
Astérisque, 137. Publié avec le concours du Centre National de la Recherche Scientifique. Paris: Société Mathématique de France. 143 p. FF 90.00; $ 13.00 (1986).

This is an interesting monograph - written in English by a German author during a research stay in America (at Brown University) and published in a French publication series. The book is an essentially self-contained beautiful treatise of the projective geometry of elliptic curves. Old and new results of other authors are combined with new results of the present author into a modern exposition using the language of algebraic geometry.

The starting point and core of the book is a paper of G. Ellingsrud and D. Laksov [18th Scand. Congr. Math., Proc., Aarhus 1980, Prog. Math. 11, 258-287 (1981; Zbl 0479.14012)] on the classification of normal bundles of elliptic space curves of fifth degree. This is now presented in a new and more general setting in chapter VIII. We summarize the author’s outline of the contents of the essential chapters of the monograph.

The projective geometry of elliptic curves C is based on the embedding via theta-functions of C as a linearly normal curve \(C_ n\) of degree \(n\) in the (n-1)-dimensional projective space \({\mathbb{P}}_{n-1}\) over the complex numbers \({\mathbb{C}}\). The embedding can be described by virtue of the operation of the Heisenberg group \(H_ n\) on \(C_ n\) via translation by n-division points. In chapter I the classical theory of the symmetries of \(C_ n\subseteq {\mathbb{P}}_{n-1}\) going back to L. Bianchi and A. Hurwitz is treated.

Chapter II generalizes to prime degrees \(n=p\geq 3\) the classical configuration of points of inflection of a plane cubic in the case of \(n=3\). This was essentially known already to C. Segre, but an independent treatment is given here.

The results of chapter I on symmetries ar applied in chapter IV for finding quadratic equations of an elliptic normal curve \(C_ n\subseteq {\mathbb{P}}_{n-1}\). This is possible because, for \(n\geq 4\), \(C_ n\) turn out to be the scheme-theoretic intersection of quadrics of rank 3, a result which follows from a more general theorem of Mumford on quadratic equations defining an abelian variety. The author obtains a new and simple proof of Mumford’s theorem for elliptic curves. Furthermore, the singular quadrics through a given elliptic normal curve \(C_ 5\subseteq {\mathbb{P}}_ 4\) are considered. It is shown that, for a one-dimensional family of rank 3 quadrics through \(C_ 5\), their singular lines form a ruled surface F of degree \(15.\)

The study of the normal bundle \(N_{C_ 5}\) of an elliptic normal quintic \(C_ 5\subseteq {\mathbb{P}}_ 4\) in chapter V leads via Atiyah’s classification of vector bundles over elliptic curves to a new proof of a vanishing theorem of Ellingsrud and Laksov which will be used in chapter VIII. The space \(\Gamma\) (\({\mathcal O}_{{\mathbb{P}}_{n-1}}(n))\) of homogeneous forms of degree \(n\) in n variables, on which the Heisenberg group \(H_ n\) operates, is discussed in chapter VI and the dimension of the subspace \(\Gamma_{H_ n}({\mathcal O}_{{\mathbb{P}}_{n-1}}(n))\) of forms invariant under \(H_ n\) determined for prime degrees \(n=p\geq 3\). In fact, \[ \dim \Gamma_{H_ p}({\mathcal O}_{{\mathbb{P}}_{n- 1}}(p))=\frac{1}{p^ 2}\left( \begin{matrix} 2p-1\\ p\end{matrix} \right)+\frac{1}{p^ 2}(p^ 2-1). \] For \(p=5\), one obtains dim \(\Gamma\) \({}_{H_ 5}({\mathcal O}_{{\mathbb{P}}_ 4}(5))=6\), a result that was first proved by Horrocks and Mumford in connection with their investigation of the Horrocks-Mumford bundle. Furthermore, the configuration of chapter II leads to a basis of \(\Gamma_{H_ 5}({\mathcal O}_{{\mathbb{P}}_ 4}(5))\) consisting of the six fundamental pentahedra introduced there. Finally, for applications in chapter VIII, a basis is constructed of the 3-dimensional space \(\Gamma_{H_ 5}(J^ 2_{C_ 5}(5))\) of invariant quintic forms which are singular along \(C_ 5\), where \(J_{C_ 5}\subseteq {\mathcal O}_{{\mathbb{P}}_ 4}\) is the sheaf of ideals of \(C_ 5\subseteq {\mathbb{P}}_ 4.\)

In chapter VII elliptic normal curves \(C_ 5\subseteq {\mathbb{P}}_ 4\) are tied up with the Horrocks-Mumford bundle on \({\mathbb{P}}_ 4\). In fact the Horrocks-Mumford bundle can be reconstructed from the tangent developable of \(C_ 5\) by the so-called Serre construction. For this, however, the reader is referred to the original paper.

The classification of normal bundles of elliptic space curves of degree 5 according to Ellingsrud and Laksov is achieved by a certain 1-parameter family of quintic hypersurfaces \(Y_ M\subseteq {\mathbb{P}}_ 4\). The base locus of this family is the union of the tangent surface Tan \(C_ 5\) and the ruled surface F of singular lines of the 1-dimensional family of rank 3 quadrics through \(C_ 5\) dealt with in chapter IV. The author now characterizes in chapter VIII a 2-dimensional space \(U\subseteq \Gamma ({\mathcal O}_{{\mathbb{P}}_ 4}(5))\) belonging to the \(Y_ M\) whose elements are invariant under the Heisenberg group \(H_ 5\). More precisely, U is made up of those \(H_ 5\)-invariant quintic forms which vanish on Tan \(C_ 5\) and whose associated hypersurfaces are singular along \(C_ 5\). In formulas, \(U=\Gamma_{H_ 5}(J_{Tan C_ 5}(5))\cap \Gamma_{H_ 5}(J^ 2_{C_ 5}(5)).\)- This yields a new description of the Horrocks-Mumford bundle. By using the basis of \(\Gamma_{H_ 5}(J^ 2_{C_ 5}(5))\) constructed in chapter VI, the space U can be explicitly described also as a subspace of \(\Gamma_{H_ 5}(J^ 2_{C_ 5}(5)).\)

In this way, among other things, a new understanding of the Ellingsrud- Laksov classification of elliptic normal curves is gained and their numerous connections with the Horrocks-Mumford bundle are explained. To combine these various aspects of the projective geometry of elliptic curves was one of the main purposes of the book.

Except for the relation between the curves \(C_ 5\subseteq {\mathbb{P}}_ 4\) and Shioda’s modular surface S(5) in chapter IV and the Serre construction in chapter VII, all results are completely proved. Moreover, new proofs of some essentially known results are supplied, such as the Segre configuration in chapter II, Mumford’s theorem on quadratic equations of abelian varieties in chapter IV, and the assertion of Horrocks and Mumford on the degeneration of a certain abelian surface which was made more precise in chapter VII. The monograph contains also some interesting new results and generalizations of old results.

The starting point and core of the book is a paper of G. Ellingsrud and D. Laksov [18th Scand. Congr. Math., Proc., Aarhus 1980, Prog. Math. 11, 258-287 (1981; Zbl 0479.14012)] on the classification of normal bundles of elliptic space curves of fifth degree. This is now presented in a new and more general setting in chapter VIII. We summarize the author’s outline of the contents of the essential chapters of the monograph.

The projective geometry of elliptic curves C is based on the embedding via theta-functions of C as a linearly normal curve \(C_ n\) of degree \(n\) in the (n-1)-dimensional projective space \({\mathbb{P}}_{n-1}\) over the complex numbers \({\mathbb{C}}\). The embedding can be described by virtue of the operation of the Heisenberg group \(H_ n\) on \(C_ n\) via translation by n-division points. In chapter I the classical theory of the symmetries of \(C_ n\subseteq {\mathbb{P}}_{n-1}\) going back to L. Bianchi and A. Hurwitz is treated.

Chapter II generalizes to prime degrees \(n=p\geq 3\) the classical configuration of points of inflection of a plane cubic in the case of \(n=3\). This was essentially known already to C. Segre, but an independent treatment is given here.

The results of chapter I on symmetries ar applied in chapter IV for finding quadratic equations of an elliptic normal curve \(C_ n\subseteq {\mathbb{P}}_{n-1}\). This is possible because, for \(n\geq 4\), \(C_ n\) turn out to be the scheme-theoretic intersection of quadrics of rank 3, a result which follows from a more general theorem of Mumford on quadratic equations defining an abelian variety. The author obtains a new and simple proof of Mumford’s theorem for elliptic curves. Furthermore, the singular quadrics through a given elliptic normal curve \(C_ 5\subseteq {\mathbb{P}}_ 4\) are considered. It is shown that, for a one-dimensional family of rank 3 quadrics through \(C_ 5\), their singular lines form a ruled surface F of degree \(15.\)

The study of the normal bundle \(N_{C_ 5}\) of an elliptic normal quintic \(C_ 5\subseteq {\mathbb{P}}_ 4\) in chapter V leads via Atiyah’s classification of vector bundles over elliptic curves to a new proof of a vanishing theorem of Ellingsrud and Laksov which will be used in chapter VIII. The space \(\Gamma\) (\({\mathcal O}_{{\mathbb{P}}_{n-1}}(n))\) of homogeneous forms of degree \(n\) in n variables, on which the Heisenberg group \(H_ n\) operates, is discussed in chapter VI and the dimension of the subspace \(\Gamma_{H_ n}({\mathcal O}_{{\mathbb{P}}_{n-1}}(n))\) of forms invariant under \(H_ n\) determined for prime degrees \(n=p\geq 3\). In fact, \[ \dim \Gamma_{H_ p}({\mathcal O}_{{\mathbb{P}}_{n- 1}}(p))=\frac{1}{p^ 2}\left( \begin{matrix} 2p-1\\ p\end{matrix} \right)+\frac{1}{p^ 2}(p^ 2-1). \] For \(p=5\), one obtains dim \(\Gamma\) \({}_{H_ 5}({\mathcal O}_{{\mathbb{P}}_ 4}(5))=6\), a result that was first proved by Horrocks and Mumford in connection with their investigation of the Horrocks-Mumford bundle. Furthermore, the configuration of chapter II leads to a basis of \(\Gamma_{H_ 5}({\mathcal O}_{{\mathbb{P}}_ 4}(5))\) consisting of the six fundamental pentahedra introduced there. Finally, for applications in chapter VIII, a basis is constructed of the 3-dimensional space \(\Gamma_{H_ 5}(J^ 2_{C_ 5}(5))\) of invariant quintic forms which are singular along \(C_ 5\), where \(J_{C_ 5}\subseteq {\mathcal O}_{{\mathbb{P}}_ 4}\) is the sheaf of ideals of \(C_ 5\subseteq {\mathbb{P}}_ 4.\)

In chapter VII elliptic normal curves \(C_ 5\subseteq {\mathbb{P}}_ 4\) are tied up with the Horrocks-Mumford bundle on \({\mathbb{P}}_ 4\). In fact the Horrocks-Mumford bundle can be reconstructed from the tangent developable of \(C_ 5\) by the so-called Serre construction. For this, however, the reader is referred to the original paper.

The classification of normal bundles of elliptic space curves of degree 5 according to Ellingsrud and Laksov is achieved by a certain 1-parameter family of quintic hypersurfaces \(Y_ M\subseteq {\mathbb{P}}_ 4\). The base locus of this family is the union of the tangent surface Tan \(C_ 5\) and the ruled surface F of singular lines of the 1-dimensional family of rank 3 quadrics through \(C_ 5\) dealt with in chapter IV. The author now characterizes in chapter VIII a 2-dimensional space \(U\subseteq \Gamma ({\mathcal O}_{{\mathbb{P}}_ 4}(5))\) belonging to the \(Y_ M\) whose elements are invariant under the Heisenberg group \(H_ 5\). More precisely, U is made up of those \(H_ 5\)-invariant quintic forms which vanish on Tan \(C_ 5\) and whose associated hypersurfaces are singular along \(C_ 5\). In formulas, \(U=\Gamma_{H_ 5}(J_{Tan C_ 5}(5))\cap \Gamma_{H_ 5}(J^ 2_{C_ 5}(5)).\)- This yields a new description of the Horrocks-Mumford bundle. By using the basis of \(\Gamma_{H_ 5}(J^ 2_{C_ 5}(5))\) constructed in chapter VI, the space U can be explicitly described also as a subspace of \(\Gamma_{H_ 5}(J^ 2_{C_ 5}(5)).\)

In this way, among other things, a new understanding of the Ellingsrud- Laksov classification of elliptic normal curves is gained and their numerous connections with the Horrocks-Mumford bundle are explained. To combine these various aspects of the projective geometry of elliptic curves was one of the main purposes of the book.

Except for the relation between the curves \(C_ 5\subseteq {\mathbb{P}}_ 4\) and Shioda’s modular surface S(5) in chapter IV and the Serre construction in chapter VII, all results are completely proved. Moreover, new proofs of some essentially known results are supplied, such as the Segre configuration in chapter II, Mumford’s theorem on quadratic equations of abelian varieties in chapter IV, and the assertion of Horrocks and Mumford on the degeneration of a certain abelian surface which was made more precise in chapter VII. The monograph contains also some interesting new results and generalizations of old results.

Reviewer: H.G.Zimmer

##### MSC:

14H10 | Families, moduli of curves (algebraic) |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |

14K25 | Theta functions and abelian varieties |

14N05 | Projective techniques in algebraic geometry |