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 MaxNoether (1876).

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, 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.