In
algebraic geometry, 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 surfaces 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
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook i ...
. Weil references Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 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 ...
'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 differenti ...
. 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, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbe ...
s, an extension of the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
line, two points are called infinitely near if their difference is
infinitesimal.
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 all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with th ...
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
:
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
:
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