Abstract
sense the zeros and poles of the form respectively. These modules are finite dimensional vector spaces, and we prove in $1 that the difference in dimension is preserved under deformation of both the form and the curve. In the case that r = dg for some holomorphic function g, this enables us to find the number of critical points of a small generic deformation ofg. Further, we give a simple proof of the fact that the jump in Milnor number (as defined in [3] and [6]) in a flat family of curve singularities is equal to the vanishing Euler characteristic. In the case that V and a are real, the l-form defines an orientation on each connected component of V - {p} ( = half-branch), where p is the base point of %, Some of these halfbranches will be oriented outwards and some inwards. Moreover, the two ramification modules come with real valued non-degenerate quadratic forms. We show in $2 that the sum of signatures of these two forms is equal to the difference between the numbers of branches oriented outwards and those oriented inwards. This is related to the classical method of Hermite for calculating the number of real roots of a polynomial as the signature of a quadratic form (see[12]). We remark that it seems surprising that the two important features of these ramification modules are the difference of the dimensions, but the sum of the signatures.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.