1,721,072 research outputs found
Remarks on the relations between the Italian and American schools of algebraic in the first decades of the 20th century
No full textIn this paper we give an overview of the interactions between Italian and American algebraic geometers during the first decades of the 20th century. We focus on three mathematicians—Julian L. Coolidge, Solomon Lefschetz, and Oscar Zariski—whose relations with the Italian school were quite intense. More generally, we discuss the importance of this influence in the development of algebraic geometry in the first half of the 20th centur
On Cremona contractibility
No full textWe give a constructive proof of a classical theorem which determines irreducible plane curves that are contractible to a point by a Cremona transformation. The problem of characterizing Cremona contractible (not necessarily irreducible) hypersurfaces in a projective space is in general widely open: we report on the only known result about reducible plane curves consisting of two components, due to litaka, and we discuss a couple of examples concerning plane curves with more components. Finally, we prove that all varieties of codimension at least two in a projective space are Cremona contractible to a point
On the Cremona contractibility of unions of lines in the plane
No full textWe discuss the concept of Cremona contractible plane curves, with a historical account on the development of this subject. Then we classify Cremona contractible unions of d ≥ 12 lines in the plane
The Italian school of algebraic geometry and Abel's legacy
No full textTHE ITALIAN SCHOOL OF ALGEBRAIC GEOMETRY AND
ABEL’S LEGACY
ALDO BRIGAGLIA, CIRO CILIBERTO, AND CLAUDIO PEDRINI
Introduction
Exactly one century ago, in the spring of 1902, the prestigious mathematical journal Acta Mathematica, decided to mark the centenary of Abel’s birth by dedicating three issues to the memory of this Norwegian mathematician. Some of the most influential Italian mathematician of the day were invited to contribute to these issues. Among those invited were Guido Catelnuovo, at the time the most well-known representative of the Italian school of algebraic geometry, and Federigo Enriques. Enriques, at first, accepted the invitation and, as he communicated to Castelnuovo, decided to write ... una specie di Bericht, breve, intorno ai nostri lavori, in cui mi propongo di esporre la teoria sotto l’aspetto analitico.
( ... a kind of Bericht, a short one, concerning our papers, in which I would like to expose the theory using the analytic viewpoint.)
(letter of 13/3/1902, for all citations from the correspondence between Enriques and Castelnuovo, see [BCG]).
The project was never completed. However this short citation is important for us: it tells us, in fact, that the Italian algebraic geometers, at the beginning of last century, used to look at their connections with Abel via analytic and transcendantal methods. Although never really central in the Italian school were, this attitude
was always considered as a natural counterpart of the projective-geometric aspects typical of the Italians. Furthermore, as we will see, at the time Enriques was writing, there was in Italy an important revival of analytic ideas.
In this paper we will try to make clear an interesting set of contributions which,
inside the Italian school, were, in Enriques’ sense, inspired by Abel’s analytic viewpoint.
For this we will go back to the real beginning of the italian school with Luigi Cremona, who is considered to be its founder. Then we will follow the evolution of these ideas up until the 1930’s and we will indicate how they are thought of today.
Abel, apparently, never had any contact with the Italian mathematical environment,in his short life, nor even specifically during his travels to Italy (see [St]).
Therefore, his influence on Italian mathematicians was late and indirect. However,
we believe that, despite this, it has been deep and longstanding.
In the first chapter we will give an overview of the influence of Abel’s ideas on the beginnings of the Italian school, filtered via Riemann’s viewpoint and the geometric interpretation of it by Clebsch and Gordan, and later by Brill and Noether. This chapter mainly centers around the character of Luigi Cremona, the founder of the
Italian school.
In the second chapter we will see how Abel’s influence was still active at the beginning of the XXth century and played a basic role in understanding the notion of irregularity of surfaces. The main characters, in this period, are the young Severi and the well established Caselnuovo, whose contributions we will review, indicating also their subsequent far reaching influence inside and outside the Italian school,
until A. Weil’s proof of Riemann’s hypotheses for the zeta function of an algebraic curve over a finite field.
The third chapter is devoted to Severi’s ideas on rational equivalence of 0-cycles on a surface. We will indicate how some of these ideas were related, in Severi’s mind, to Abel’s viewpoint. As is well known, Severi’s contributions on the subject have been very controversial. We will briefly report on the main criticisms but we will also try to elucidate which of them have a present validity. In particular we will direct our, and we hope the reader’s attention, to some of his ideas which are very closely related to Bloch’s conjecture. We dedicate a few, more technical, sections at the end of chapter 3 to open a window on the present developments of this last
subject, specifically to some motivic interpretations which we think are rather close
to Severi’s original viewpoint.
Due to the different tastes and attitudes of the authors, which we deliberately did not make too much effort to hide, the paper is rather uneven. The first chapter is expository and more historiographical in nature. The other two, though sharing with the first a historiographical perspective, have a different flavour: in the second some technical aspects start to appear, and they become even more relevant in chapter 3, especially, as we said, in its last sections. We hope that the uneveness
of the paper will attract, rather than repel, readers with different interests
Remarks on the minimal genus of curves linearly moving on a surface
Full text linkGiven a smooth, irreducible, projective surface , let be the
minimum geometric genus of an irreducible curve that moves in a linear system
of positive dimension on . We determine the value of this birational
invariant for a general surface of degree in and give a bound
for if is a general polarised K3 or abelian surface. As soon as this
note appeared on the math arxiv, David Stapleton kindly pointed out to the
author a paper by Ein and Lazarsfeld, in which in turn a paper by Konno was
cited. Unfortunately the author was not aware of these two papers which contain
results that strictly include the ones of this note. The author is very
grateful to David Stapleton
On complete intesections containing a linear subspace
No full textConsider the Fano scheme F_k(Y) parameterizing k-dimensional linear subspaces contained in a complete intersection Y in IP^m of multi-degree d = (d_1,....,d_s). It is known that, if t:= t(m,d,k) <= 0 and d_1....d_s >2, for Y a general complete intersection as above, then F_k(Y) has dimension −t. In this paper we consider the case t>0. Then the locus W(d,k) of all complete intersections as above containing a k-dimensional linear subspace is irreducible and turns out to have codimension t in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general [Y] in W(d,k) the scheme F_k(Y) is zero-dimensional of length one. This implies that W(d,k) is rational
Cones of lines having high contact with general hypersurfaces and applications
No full textGiven a smooth hypersurface of degree , we study the cones swept out by lines
having contact order at a point . In particular, we
prove that if is general, then for any and , the cone has dimension exactly . Moreover, when
is a very general hypersurface of degree , we describe the
relation between the cones and the degree of irrationality of
--dimensional subvarieties of passing through a general point of . As
an application, we give some bounds on the least degree of irrationality of
--dimensional subvarieties of passing through a general point of ,
and we prove that the connecting gonality of satisfies
Corrigendum to the paper "On the K^2 of degenerations of surfaces and the multiple point formula"
No full textWe correct an error in the Multiple Point Formula (7.3) in the paper mentioned in the title. This correction propagates to formulas (7.5), (7.6), (7.23) and (8.18), and it affects minor results in Section 8, where few statements require an extra assumption, but it does not affect the main results of Section 8
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