Plane curves with small linear orbits, I

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...

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 ...

An Index Calculus Algorithm for Plane Curves of Small Degree

We present an index calculus algorithm which is particularly well suited to solve the discrete logarithm problem (DLP) in degree 0 class groups of curves over finite fields which are represented by plane models of small degree. A heuristic analysis of our algorithm indicates that asymptotically for varying q, “almost all” instances of the DLP in degree 0 class groups of curves represented by pl...

On the Intersections of Rational Curves with Cubic Plane Curves

Let V be a generic quintic threefold in the 4-dimensional complex projective space P. A well-known conjecture of Clemens says that V has only a finite number of rational curves in each degree. Although Clemens’ conjecture is still quite open at this moment (it is known to be true for degree up to 7 by S. Katz [K]), recently physicists, based on the mirror symmetry principle, have proposed a for...