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Publication# Examples of normal, not geometrically normal, projective Gorenstein del Pezzo surfaces with at most Du Val singularities

Abstract

While over fields of characteristic at least 5, a normal, projective and Gorenstein del Pezzo surface is geometrically normal, this does not hold for characteristic 2 and 3. There is no characterization of all such non-geometrically normal surfaces, but there is a complete characterization of all possible surfaces that can arise as the normalization of the base change to the algebraic closure of the base field. In four of these instances, for a given normalization, we describe the construction of such a non-geometrically normal surface.

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Related concepts (32)

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Normal (geometry)

In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one (in which case it is a unit normal vector) or its length may represent the curvature of the object (a ); its algebraic sign may indicate sides (interior or exterior).

Log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values.

Implicit surface

In mathematics, an implicit surface is a surface in Euclidean space defined by an equation An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for x or y or z. The graph of a function is usually described by an equation and is called an explicit representation. The third essential description of a surface is the parametric one: where the x-, y- and z-coordinates of surface points are represented by three functions depending on common parameters .

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