Regularity of linear systems of plane curves

Linear Systems of Plane Curves

Non-trivial Linear Systems on Smooth Plane Curves

Let C be a smooth plane curve of degree d defined over an algebraically closed field k. In [10], while studying space curves, Max Noether considered the following question. For n ∈ Z≥1 find l(n) ∈ Z≥0 such that there exists a linear system g l(n) n on C but no linear system g l(n)+1 n and classify those linear systems g l(n) n on C. The arguments given by Noether in the answer to this question ...

Approximate parametrization of plane algebraic curves by linear systems of curves

It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance ǫ > 0 and an ǫ-irreducible algebraic affine plane curve C of proper degree d, we introduce the notion of ǫ-rationality, and we provide an algorithm to parametrize approximately affine ǫ-rational plane curves, without exact singularities at infinit...

Linear Orbits of Arbitrary Plane Curves

The `linear orbit' of a plane curve of degree d is its orbit in P under the natural action of PGL(3). In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane curve, and give explicit formulas for plane curves with irreducible singularities. The main tool is an intersection-theoretic study of the projective normal cone of a scheme determ...

Application of the linear Differential Equations on the Plane and Elements of Nonlinear Systems, In Economics

In recent years, it has become increasingly important to incorporate explicit dynamics in economic analysis. These two tools that mathematicians have developed, differential equations and optimal control theory, are probably the most basic for economists to analyze dynamic problems. In this paper I will consider the linear differential equations on the plane (phase diagram) and elements of nonl...