Pierre Lairez

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Équipe Specfun

email
INRIA Saclay Île-de-France, Bât. Alan Turing
1 rue Honoré d'Estienne d'Orves
Campus de l'École polytechnique
91120 Palaiseau
France
location
48.7146, 2.2056
map
phone
(0033) 01 77 57 80 36
office
1155
gpg public key

## Latest work

### A numerical transcendental method in algebraic geometry

2018

In “A transcendental method in algebraic geometry”, Griffiths emphasized the role of certain multivariate integrals, known as periods, “to construct a continuous invariant of arbitrary smooth projective varieties”. Periods often determine the projective variety completely and therefore its algebraic invariants. Translating periods into discrete algebraic invariants is a difficult problem, exemplified by the long standing Hodge conjecture which describes how periods determine the algebraic cycles within a projective variety.

Recent progress in computer algebra makes it possible to compute periods with high precision and put transcendental methods into practice. We focus mainly on algebraic surfaces and give a numerical method to compute Picard groups. As an application, we count smooth rational curves on quartic surfaces using the Picard group.

$${x^4 + y^3 z + xyzw + z^3 w + yw^3 = 0} \subset \mathbb{P}^3$$

A K3 surface containing 133056 smooth rational curves of degree 4 generating a Picard group of rank 19.