Clearly, one can easily give an upper bound for the number of
singular points that may occur on a plane irreducible curve of degree :
by the genus formula can have, at most,
singularities. Another upper bound for the (weighted) number of singularities
applying Bézout's Theorem to the intersection of two generic polars of :
On the other hand, in the case of arbitrary topological types , we have the following existence theorem, which is asymptotically optimal (with respect to the occurring invariants and the exponent of )
Theorem: [GLS,Lo]. if and two additional conditions for the five ``worst'' singularities hold true.
In case of only one singularity we have the slightly better sufficient condition for existence, .
The theorem is just an existence statement, the proof gives no hint how to produce any equation. To produce explicit equations one needs some constructive method. Then the computer can be used in order to check the construction, or even, to improve the results. The following is a prominent example (actually, it belongs to a series of ``world record'' examples):
Example: [GN] The irreducible curve with affine equation
In order to verify this, one may proceed, using SINGULAR, as follows:
> ring s = 0,(x,y),ds; > poly f = y2-2x28y-4x21y17+4x14y33-8x7y49+20y65+x56+4x49y16; > matrix Hess = jacob(jacob(f)); //the Hessian matrix of f > vdim(std(jacob(f))); //the Milnor number of f 2260Since the rank of the Hessian at 0 is checked to be 1, has an singularity at 0; it is an -singularity since the Milnor number is 2260. In the following we show that the projective curve defined by has no further singularities in the affine part. This follows from
> vdim(std(jacob(f)+f)); 2260 // multiplicity of Sing(C) at 0 (local ordering) > ring r = 0,(x,y),dp; > poly f = fetch(s,f); > vdim(std(jacob(f)+f)); 2260 // multiplicity of Sing(C) (global ordering)Finally, we have to consider the singularities at infinity:
> ring sh = 0,(x,y,z),dp; > poly f = fetch(s,f); > poly F = homog(f,z); F; // homogeneous polynomial defining C 4x49y16+20y65+x56z9-8x7y49z9+4x14y33z18-4x21y17z27-2x28yz36+y2z63 > ring r1 = 0,(y,z),dp; > map phi = sh,1,y,z; > poly g = phi(F); // F in affine chart (x=1) > vdim(std(jacob(g)+g)); 120 > ring r2 = 0,(y,z),ds; // local ring at (1:0:0) > poly g = fetch(r1,g); g; z9+4y16-2yz36-4y17z27+4y33z18-8y49z9+20y65+y2z63 > vdim(std(jacob(g)+g)); 120As before, we can conclude that there is precisely one singularity of on the line at infinity, situated at , being semiquasihomogeneous of type . (Note that in our computation we have considered all points at infinity except . The latter is obviously not a point of ).
In the following we should like to mention a few problems and conjectures which are currently in the centre of research in connection with singular curves in .
Computing zero-dimensional ideals
Many of the questions concerning plane projective curves with prescribed singularities can be translated to properties of zero-dimensional (homogeneous) ideals , e.g.,
Harbourne-Hirschowitz conjecture: Let , in general position, a positive integer, and let . Then the Hilbert function satisfies
There are several special cases where this conjecture is known to hold true; in particular, C. Ciliberto and R. Miranda [CM] have proven that it always holds for . Nevertheless the general conjecture is still far from being proven.
Nagata conjecture: Let , in general position, positive integers, and let denote the minimal degree of a curve passing through each of the points with multiplicity (at least) , . Then
Computer algebra could be used to provide evidence for such conjectures (or, to produce counter examples) provided one can solve the following problems: