Contact Unimodular Germs From the Plane to the Plane

Volume preserving subgroups of $${{\mathcal A}}$$ and $${{\mathcal K}}$$ and singularities in unimodular geometry

“…(and non-w.q.h.) map-germs f from the classifications in [14,15,51]. These results give upper bounds for dim M (K n , f ), and for certain f some of these upper bound will coincide with the following lower bound (which is analogous to Lemma 5.1 in the A p case).…”

“…And for each f λ = (xz + x y 2 + y 3 , yz, x 2 + y 3 + λz q ), λ ∈ C, the family f λ a = (xz + x y 2 + y 3 , yz, x 2 + ay 3 + λz q ) parameterizes the K n -orbits inside K · f λ . Example 6.4 Finally, consider the K-unimodal equidimensional maps of type G k,l,m from [14], given by f = (g 1 , g 2 ) = (x 2 + y k , x y l + y m ) = f 0 + (0, y m ), where k = 2(m −l) and either k ≤ l, l +1 < m < l +k −1 (case (a)) or l < k < 2l −1, k < m < 2l (case (b)). As above we check that the coefficient of (0, y m ) is a K n -modulus, hence dim M(K n , f ) ≥ 1.…”

Contact unimodal map germs from the plane to the plane

“…, f p of f . Table 1 contains all map germs from the plane to the plane of Boardman symbol (2,2) given by Dimca and Gibson [9].…”

“…The computation shows that (x 3 + y 4 , x 2 y + y 4 ), denoted by X 1 is contact equivalent to g = (x 3 + x y 3 + y 4 , x 2 y + y 4 )+ terms of order ≥ 5. g has 4-jet (x 3 + x y 3 + y 4 , x 2 y + y 4 ), denoted by Q 4 . In the paper of Dimca and Gibson [9] it is proved (Lemma 3.6) that g is contact equivalent to its 4-jet (x 3 + x y 3 + y 4 , x 2 y + y 4 ). This implies that X 1 is contact equivalent to Q 4 .…”

“…These classifications are characterized in [3,5,11] in terms of certain invariants. In [9], Dimca and Gibson gave the classification of all map germs from the plane to the plane of Boardman symbol (2,1) and (2,2) with respect to K -equivalence. A classifier for the classification of Dimca and Gibson, in the case of Boardman symbol (2, 1) is given in [4].…”