Search results for "Double point"

Double point curves for corank 2 map germs from C2 to C3

2012

Abstract We characterize finite determinacy of map germs f : ( C 2 , 0 ) → ( C 3 , 0 ) in terms of the Milnor number μ ( D ( f ) ) of the double point curve D ( f ) in ( C 2 , 0 ) and we provide an explicit description of the double point scheme in terms of elementary symmetric functions. Also we prove that the Whitney equisingularity of 1-parameter families of map germs f t : ( C 2 , 0 ) → ( C 3 , 0 ) is equivalent to the constancy of both μ ( D ( f t ) ) and μ ( f t ( C 2 ) ∩ H ) with respect to t , where H ⊂ C 3 is a generic plane.

The doodle of a finitely determined map germ from R2 to R3

AbstractLet f:U⊂R2→R3 be a representative of a finitely determined map germ f:(R2,0)→(R3,0). Consider the curve obtained as the intersection of the image of the mapping f with a sufficiently small sphere Sϵ2 centered at the origin in R3, call this curve the associated doodle of the map germ f. For a large class of map germs the associated doodle has many transversal self-intersections. The topological classification of such map germs is considered from the point of view of the associated doodles.

Double points in families of map germs from ℝ2 to ℝ3

2020

We show that a 1-parameter family of real analytic map germs [Formula: see text] with isolated instability is topologically trivial if it is excellent and the family of double point curves [Formula: see text] in [Formula: see text] is topologically trivial. In particular, we deduce that [Formula: see text] is topologically trivial when the Milnor number [Formula: see text] is constant.

Luigi Cremona’s Years in Bologna: From Research to Social Commitment

2011

Luigi Cremona (1830–1903), unanimously considered to be the man who laid the foundations of the prestigious Italian school of Algebraic Geometry, was active at the University of Bologna from October 1860, when assigned by the Minister Terenzio Mamiani (1799–1885) to cover the Chair of Higher Geometry, until September 1867 when Francesco Brioschi (1824–1897) called him to the Politecnico di Milano. The “Bolognese years” were Cremona’s richest and most significant in terms of scientific production, and, at the same time, were the years when he puts the basis for its most important interventions in the social and political life of the “newborn” kingdom of Italy. In this article we present thes…