Acnodes

An acnode is a type of isolated singular point on an algebraic curve. It is a point where the curve is self-intersecting, but unlike a crunode, the curve does not cross itself at an acnode. Instead, the curve touches itself at this point, forming a sharp cusp. More formally, an acnode is a singular point that is also a node.

In algebraic geometry, the classification of singular points is important for understanding the geometry of curves. Singular points are points where the curve is not smooth, meaning it does not have a well-defined tangent line. Acnodes are characterized by specific conditions on the partial derivatives of the curve's defining polynomial. For a curve defined by the equation f(x, y) = 0, a point (a, b) is an acnode if the partial derivatives of f with respect to x and y are both zero at (a, b), and if the Hessian matrix of f at (a, b) has a positive determinant. This condition indicates that the point is a local minimum for the function f, which corresponds to the curve touching itself at that point.

Acnodes are relatively rare compared to other types of singular points like ordinary double points or cusps. They represent a specific degeneracy in the geometric structure of an algebraic curve. The study of acnodes helps in classifying and understanding the global properties of algebraic curves, particularly in fields like algebraic topology and differential geometry.