Infinitely near point

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 Max Noether (1876).[1]

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,[2] though these are not the same as infinitely near points in algebraic geometry. In the line of hyperreal numbers, an extension of the real number line, two points are called infinitely near if their difference is infinitesimal.

Definition

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 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 S₀ is given by a sequence of points P₀, P₁,...,P_n on surfaces S₀, S₁,...,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 P₀, P₁,... 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 _{x{\text{ infinitely near }}p}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 = x² and p = (0,0).

The genus of C is given by

g(C)=g(N)+\sum _{{\text{infinitely near points }}x}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

Infinitely Near Points on Algebraic Surfaces, Gino Turrin, American Journal of Mathematics, Vol. 74, No. 1 (Jan., 1952), pp. 100–106
[4] 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'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 here) for a full discussion.

Noether, M. (1876), "Ueber die singularen Werthsysteme einer algebraischen Function und die singularen Punkte einer algebraischen Curve", Mathematische Annalen, 9 (2): 166–182, doi:10.1007/BF01443372, S2CID 120376948

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[1] Infinitely Near Points on Algebraic Surfaces, Gino Turrin, American Journal of Mathematics, Vol. 74, No. 1 (Jan., 1952), pp. 100–106

[2] [4] 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'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 here) for a full discussion.