Infinitely near point

In mathematics, the notion of infinitely near points was initially part of the intuitive foundations of differential calculus. In the simplest terms, two points which lie at an infinitesimal distance apart are considered infinitely near.

Explanation

In more geometric terms, a notion of infinitely near point is a necessary tool of birational geometry, as soon as algebraic surfaces are considered, and was introduced in the nineteenth century. ["Infinitely Near Points on Algebraic Surfaces", Gino Turrin, "American Journal of Mathematics", Vol. 74, No. 1 (Jan., 1952), pp. 100-106, attributes the usage to Max Noether.] When blowing up is applied to a point "P" on a surface "S", the new surface "S"* contains a whole curve "C" where "P" used to be. The points of "C" have the geometric interpretation as the tangent directions at "C" to "S". They can be called infinitely near to C as way of visualizing them on "S", rather than "S"*. [Blowing up can be iterated. Yuri Manin, in "Cubic Forms", contemplates the limit of all blowings up, calling it a 'bubble spac'’. A less dramatic construction is that of the Zariski Riemann surface.]

A new language for Taylor polynomials was introduced from the 1930s, as the theory of jets. In 1953 André Weil wrote on the topic of infinitely near points, on smooth manifolds "M", from the point of view of commutative algebra. [ [4] Weil, A., "Theorie des points proches sur les varietes differentielles", Colloque de Topologie et Geometrie Diferentielle, Strasbourg, 1953, 111-117; in his "Collected Papers" II. The notes to the paper there indicate this was a rejected project for the Bourbaki group. Weil references Pierre de Fermat's approach to calculus, as well as the jets of Charles Ehresmann. For an extended treatment, see O. O. Luciano, "Categories of multiplicative functors and Weil's infinitely near points", Nagoya Math. J. 109 (1988), 69–89 (online [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.nmj/1118780892 here] for a full discussion.]

With the advent of scheme theory, infinitesimal neighbourhoods in algebraic geometry could be defined routinely for an algebraic variety "V" (and more generally). The diagonal embedding of "V" in "V" × "V" being defined by "I" (an ideal), use "I"² to define a first-order infinitesimal neighbourhood "N". The structure sheaf to "N" then contains nilpotents; these have no classical meaning but ensure that the scheme-theoretic points of "N" do carry first-order infinitesimal information. This construction generalizes that of the dual numbers (which constitute the affine ring of the first-order neighbourhood of a point on the affine line. [David Mumford, "Red Book" draws out these implications of Grothendieck's theory.]

Notes