A. Collino

A. Collino
Università degli Studi di Torino | UNITO · Dipartimento di Matematica "Giuseppe Peano"

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47
Publications
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422
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Introduction

Publications

Publications (47)
Article
On the second symmetric product \(C^{(2)} \) of a hyperelliptic curve \(C\) of genus \(g\) let \(L\) be the line given by the divisors on the standard linear series \(g^1_2\) and for a point \(b \in C\) let \(C_b\) be the curve \(\{(x+b) : x \in C \}\). It is proved that \(\pi _1 ( C^{(2)} \setminus (L \cup C_b) ) \) is the integer-valued Heisenber...
Article
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A collection of selected papers of Gino Fano, containing most of his research papers in Italian, is now available on line at the site of Biblioteca Digitale Italiana di Matematica. The address is: http://www.bdim.eu/ The realization of this collection on line is part of a joint project of Unione Matematica Italiana (UMI) and Societa' Italiana per l...
Article
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We analize the semistable degeneration of the Fano surface F when the cubic threefold becomes the Segre primal. This gives an explicit topological decomposition for F. The decomposition is used to decide that the Fano surface is not an an Eilenberg Mac-Lane K(\pi,1) space, this was the question that prompted us to look into the matter.
Article
The Fano surface $F$ of lines in the cubic threefold $V$ is naturally embedded in the intermediate Jacobian $J(V)$, we call "Fano cycle" the difference $F-F^-$, this is homologous to 0 in $J(V)$. We study the normal function on the moduli space which computes the Abel-Jacobi image of the Fano cycle. By means of the related infinitesimal invariant w...
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In this paper we study higher Chow groups of smooth, projective surfaces over a field k of characteristic zero, using some new Hodge theoretic methods which we develop for this purpose. In particular we investigate the subgroup of CH r+1 (X,r) with r = 1,2 consisting of cycles that are supported over a normal crossing divisor Z on X. In this case,...
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We construct some natural indecomposable elements of \(CH^g(J(C),1)\) with trivial regulator, and in particular, prove that \(\) is uncountable for C a generic curve or a generic hyperelliptic curve of genus \(g \geq 3\).
Article
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We study the higher Chow groups $CH^2(X,1)$ and $CH^3(X,2)$ of smooth, projective algebraic surfaces over a field of char 0. We develop a theoretical framework to study them by using so-called higher normal functions and higher infinitesimal invariants when the cycles are supported on normal crossing divisors. Then we investigate their structure on...
Article
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Title: Indecomposable Higher Chow Cycles on Low Dimensional Jacobians Authors: Alberto Collino Comments: AMS-TeX, 10 pages Subj-class: Algebraic Geometry MSC-class: 14C30 ;19E15 There is a basic indecomposable higher cycle K in Bloch's higher Chow group CH^{g}(J(C),1) on the Jacobian J(C) of a general hyperelliptic curve C of genus g. Consider K(t)...
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We give an explicit procedure which computes for degree d≤ 3 the correlation functions of topological sigma model (A-model) on a projective Fano hypersurface X as homogeneous polynomials of degree d in the correlation functions of degree 1 (number of lines). We extend this formalism to the case of Calabi–Yau hypersurfaces and explain how the polyno...
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We prove that the Griffiths group of 3-cycles homologous to zero modulo algebraic equivalence, on a generic hypersurfaces of dimension 7 and degree 3 is not finitely generated, even when tensored with Q. Using this and a result of Nori, we give examples of varieties for which some Griffiths group is not finitely generated (modulo torsion) but whose...
Article
We give a simple proof of the classical theorem of Torelli, based on Torelli's original approach and on the use of Poincaré's formulas.
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We give a simple proof of the classical theorem of Torelli, based on Torelli’s original approach and on the use of Poincaré’s formulas.
Article
Let X be a general cubic fivefold, JX the associated intermediate Jacobian, F the Fano surface of the planes contained in X. We prove that the Abel-Jacobi map induces an isomorphism from the Albanese variety of F to JX.
Article
We give a characteristic free proof of a criterion which was first proved by Ran over C.
Article
We give a characteristic free proof of a criterion which was first proved by Ran over C.
Article
Let F ⊆ P4 be a 3-fold with one ordinary double point p, and let F′ be the proper transform of F under the blowing up of P4 at p. If H ⊆ F′ is the preimage of p on F′ we prove that for F general the algebraic 1-cycle given by the difference of the two generators of the smooth quadric surface H, is not algebraically equivalent to zero on F′ Griffith...

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