Infinitely Near Point
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In
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, an infinitely near point of an algebraic surface ''S'' is a point on a surface obtained from ''S'' by repeatedly blowing up points. Infinitely near points of
algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s were introduced by . There are some other meanings of "infinitely near point". Infinitely near points can also be defined for higher-dimensional varieties: there are several inequivalent ways to do this, depending on what one is allowed to blow up. Weil gave a definition of infinitely near points of smooth varieties, Weil, A., ''Theorie des points proches sur les variétés 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 Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
's approach to calculus, as well as the jets of
Charles Ehresmann Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory. He was an early member of the Bourbaki group, and is known for his work on the differentia ...
. 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 (onlin
here
for a full discussion.
though these are not the same as infinitely near points in algebraic geometry. In the line of
hyperreal number In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer n
s, an extension of the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
line, two points are called infinitely near if their difference is
infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
.


Definition

When
blowing up In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with the space of all directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the poin ...
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 ''P'' to ''S''. They can be called infinitely near to ''P'' as way of visualizing them on ''S'', rather than ''S''*. More generally this construction can be iterated by blowing up a point on the new curve ''C'', and so on. An infinitely near point (of order ''n'') ''P''''n'' on a surface ''S''0 is given by a sequence of points ''P''0, ''P''1,...,''P''''n'' on surfaces ''S''0, ''S''1,...,''S''''n'' such that ''S''''i'' is given by blowing up ''S''''i''–1 at the point ''P''''i''–1 and ''P''i is a point of the surface ''S''i with image ''P''''i''–1. In particular the points of the surface ''S'' are the infinitely near points on ''S'' of order 0. Infinitely near points correspond to 1-dimensional valuations of the function field of ''S'' with 0-dimensional center, and in particular correspond to some of the points of the Zariski–Riemann surface. (The 1-dimensional valuations with 1-dimensional center correspond to irreducible curves of ''S''.) It is also possible to iterate the construction infinitely often, producing an infinite sequence ''P''0, ''P''1,... of infinitely near points. These infinite sequences correspond to the 0-dimensional valuations of the function field of the surface, which correspond to the "0-dimensional" points of the Zariski–Riemann surface.


Applications

If ''C'' and ''D'' are distinct irreducible curves on a smooth surface ''S'' intersecting at a point ''p'', then the multiplicity of their intersection at ''p'' is given by :\sum_ m_x(C)m_x(D) where ''m''''x''(''C'') is the multiplicity of ''C'' at ''x''. In general this is larger than ''m''''p''(''C'')''m''''p''(''D'') if ''C'' and ''D'' have a common tangent line at ''x'' so that they also intersect at infinitely near points of order greater than 0, for example if ''C'' is the line ''y'' = 0 and ''D'' is the parabola ''y'' = ''x''2 and ''p'' = (0,0). The genus of ''C'' is given by : g(C)=g(N)+\sum_m_x(m_x-1)/2 where ''N'' is the normalization of ''C'' and ''m''''x'' is the multiplicity of the infinitely near point ''x'' on ''C''.


References

*{{citation, first=M. , last=Noether, title=Ueber die singularen Werthsysteme einer algebraischen Function und die singularen Punkte einer algebraischen Curve, journal= Mathematische Annalen , volume= 9 , year=1876, issue=2 , pages=166–182, doi=10.1007/BF01443372, s2cid=120376948 Geometry Differential calculus Nonstandard analysis Birational geometry