Andrea Del Centina

Weierstrass points and their impact in the study of algebraic curves: a historical account from the “Lückensatz” to the 1970s

In this note we give a historical account of the origin and the development of the concept of Weierstrass point. We also explain how Weierstrass points have contributed to the study of compact Riemann surfaces and algebraic curves in the century from Weierstrass’ statement of the gap theorem to the 1970s. In particular, we focus on the seminal work of Hürwitz that raised questions which are at the center of contemporary research on this topic.

Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem

Published here, and discussed, are some manuscripts and a letter of Sophie Germain concerning her work on Fermat’s Last theorem. These autographs, held at Bibliothèque Nationale of Paris, at the Moreniana Library of Florence and at the University Library of Göttingen, contribute to a substantial revaluation of her work on this subject.

Projective surfaces with bi-elliptic hyperplane sections

We study projective surfaces X which have a bi-elliptic curve (i.e. 2∶1 covering of an elliptic curve) among their hyperplane sections . We give a complete characterization of those surfaces when their degree d is d≥17 (only conic bundles and scrolls if d≥19, three possible exception otherwise) and when d≤8. A conjecture is given for the remaining cases. The main tool we use is the study of the adjunction mapping on X.

Poncelet’s porism: a long story of renewed discoveries, I

In 1813, J.-V. Poncelet discovered that if there exists a polygon of n-sides, which is inscribed in a given conic and circumscribed about another conic, then infinitely many such polygons exist. This theorem became known as Poncelet’s porism, and the related polygons were called Poncelet’s polygons. In this article, we trace the history of the research about the existence of such polygons, from the “prehistorical” work of W. Chapple, of the middle of the eighteenth century, to the modern approach of P. Griffiths in the late 1970s, and beyond. For reasons of space, the article has been divided into two parts, the second of which will appear in the next issue of this journal.

On Kepler’s system of conics in Astronomiae pars optica

This is an attempt to explain Kepler’s invention of the first “non-cone-based” system of conics, and to put it into a historical perspective.

On certain remarkable curves of genus five

The aim of this note is twofold. First to show the existence of genus five curves having exactly twenty four Weierstrass points, which constitute the set of fixed points of three distinct elliptic involutions on them. Second to characterize these curves, in fact we prove that all such curves are bielliptic double cover of Fermats quartic.

Corrigendum to: "The manuscript of Abel's Parisian memoir found in its entirety" (Historia Mathematica 29 (2002) 65-69) ✩

In the note above I announced the discovery in July 2000 of the eight missing pages of the manuscriptof Abel’s Parisian work Memoire sur une propriete generale d’une classe tres etendue de fonctionstranscendantes [Abel, 1841]: precisely those numbered 21–24 and 31–34. I thought that these pages hadbeen written by Abel himself because they are not copied from the printed article (see below) and theyfit in perfectly (like a puzzle) with the other pages of the manuscript preserved at the Moreniana Libraryin Florence. Unfortunately this is not the case, and the search for the missing pages has not ended yet!I realized this only in February of 2002 after my note was printed, when by mere chance I found anotherset of pages of Abel’s manuscript, numbered 31–34, at the Labronica Library in Livorno.