In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.
A plane curve defined by an implicit equation
where F is a smooth function is said to be singular at a point if the Taylor series of F has order at least 2 at this point.
The reason for this is that, in differential calculus, the tangent at the point (x0, y0) of such a curve is defined by the equation
whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent may not be defined in the standard way, either because it does not exist or a special definition must be provided.
In general for a hypersurface
the singular points are those at which all the partial derivatives simultaneously vanish. A general algebraic variety V being defined as the common zeros of several polynomials, the condition on a point P of V to be a singular point is that the Jacobian matrix of the first order partial derivatives of the polynomials has a rank at P that is lower than the rank at other points of the variety.
Points of V that are not singular are called non-singular or regular. It is always true that almost all points are non-singular, in the sense that the non-singular points form a set that is both open and dense in the variety (for the Zariski topology, as well as for the usual topology, in the case of varieties defined over the complex numbers).
In case of a real variety (that is the set of the points with real coordinates of a variety defined by polynomials with real coefficients), the variety is a manifold near every regular point.
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In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.
En mathématiques, et plus particulièrement en géométrie, la notion de variété peut être appréhendée intuitivement comme la généralisation de la classification qui établit qu'une courbe est une variété de dimension 1 et une surface est une variété de dimension 2. Une variété de dimension n, où n désigne un entier naturel, est un espace topologique localement euclidien, c'est-à-dire dans lequel tout point appartient à une région qui s'apparente à un tel espace.
En mathématiques, une courbe cubique est une courbe algébrique plane définie par une équation du troisième degré en les coordonnées homogènes [X:Y:Z] du plan projectif ; ou bien c'est la version non homogène pour l'espace affine obtenue en faisant Z = 1 dans une telle équation. Ici F est une combinaison linéaire non nulle des monômes de degré trois X3, X2Y, ..., Z3 en X,Y et Z. Ceux-ci sont au nombre de dix ; donc les courbes cubiques forment un espace projectif de dimension 9, au-dessus de n'importe quel corps commutatif K donné.
In this seminar we will study toric varieties, a well studied class of algebraic varieties which is ubiquitous in algebraic geometry, but also relevant in theoretical physics and combinatorics.