Let be an irreducible projective curve of degree and a local equation for the germ . Let be the Milnor number of at .
Since the genus of , is non-negative (where , , and the normalisation of ), can have, at most, singularities.
It is a classical and interesting problem, which is still in the centre of theoretical research, to study the variety of (irreducible) curves of degree having exactly singularities of prescribed (topological or analytical) type . Among the most important questions are:
Severi (1921): and -smooth .
Harris (1985): is irreducible (if ).
Even for cuspidal curves a sufficient and necessary answer to any of the above questions is unknown.
Concerning arbitrary (topological types of) singularities, we have the
following existence theorem, which is, with respect to the exponent
of , asymptotically optimal.
Theorem: [Greuel, Lossen and Shustin1998,Lossen1999].
if and two additional conditions for the five ``worst'' singularities.
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. Having a method for constructing curves of low degree with many singularities, Lossen was able to produce explicit equations. In order to check his construction and improve the results, he made extensive use of SINGULAR to compute standard bases for global as well as for local orderings. One of his examples is the following:
Example: [Lossen1999] 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 = y2-2x10y-x9y3+1/4x8y5-1/8x7y7+5/64x6y9-7/128x5y11+21/512x4y13 -33/1024x3y15+429/16384x2y17+x20-715/32768xy19+x19y2+2431/131072y21; matrix Hess = jacob(jacob(f)); //the Hessian matrix of f print(subst(subst(Hess,x,0),y,0)); //the Hessian matrix for x=y=0 ==> 0,0, ==> 0,2 vdim(std(jacob(f))); //the Milnor number of f ==> 228
Since the rank of the Hessian at 0 is 1, has an singularity at 0; it is an singularity since the Milnor number is 228. To show that the projective curve defined by has no other singularities, we have to show that has no further singularities in the affine part and no singularity at infinity. The second assertion is easy, the first follows from
vdim(std(jacob(f)+f)); ==> 228 //multiplicity of Sing(C) at 0 (local ordering) ring r = 0,(x,y),dp; poly f = fetch(s,f); vdim(std(jacob(f)+f)); ==> 228 //multiplicity of Sing(C) (global ordering)