Plane curves of minimal degree with prescribed singularities

Plane Curves of Minimal Degree with Prescribed Singularities

We prove that there exists a positive α such that for any integer d ≥ 3 and any topological types S1, . . . , Sn of plane curve singularities, satisfying μ(S1) + · · ·+ μ(Sn) ≤ αd, there exists a reduced irreducible plane curve of degree d with exactly n singular points of types S1, . . . , Sn, respectively. This estimate is optimal with respect to the exponent of d. In particular, we prove tha...

On the Existence of Plane Curves with Prescribed Singularities

An asymptotic existence theorem for plane curves with prescribed singularities

Let d,m1, . . . ,mr be (r + 1) positive integers. Denote by V (d;m1, . . . ,mr) the variety of irreducible (complex) plane curves of degree d having exactly r ordinary singularities of multiplicities m1, . . . ,mr. In most cases, it is still an open problem to know whether this variety is empty or not. In this paper, we will concentrate on the case where the r singularities can be taken in a ge...

New Asymptotics for the Existence of Plane Curves with Prescribed Singularities

We study the classical problem of the existence of irreducible plane curves with given degree d and prescribed singularities of topological types S1 ; : : : ; Sr. Our main results concern substantial improvements of the leading coeecients in the asymptotically optimal suucient conditions for complex plane curves which were found in GLS1]. We then transfer these results to real plane curves.

Efficient Resolution of Singularities of Plane Curves

We give a new algorithm for resolving singularities of plane curves. The algorithm is polynomial time in the bit complexity model, does not require factorization, and works over Q or finite fields.