Construction of arrangements of an M-quartic and an M-cubic
with a maximal intersection of an oval and the odd branch
by S.Yu. Orevkov
-- Ca c'est vrai, dit le petit prince.
-- Et qu'en fais-tu? -- Je les gére.
Je les compte et je les recompte, dit le businessman. C'est difficile.
Mais je suis un homme sérieux!
Antoine de Saint-Exupery. "Petit Prince".
Abstract: We construct
237 pairwise non-isotope mutual arrangements of an M-quatric and
an M-cubic on RP² such that an
oval of the quartic meets the odd branch of the cubic at 12 point. Some
restrictions for such arrangements are proved. In Figure 13 we present
an arrangement wich is realizable by real pseudo-holomorphic quartic and
cubic, but which is algebraically unrealizable. The proof is based on the
cubic resolvent of a polynomial of degree 4.
All the 237 arrangements are explicitly depicted in Section 5.4
(pages 28-33). One can see the main result without reading any word. The
only thing that one should know is that we do not depict the ovals of the
quartic which are "at the infinity", i.e. in the component of
RP²
- (O4 U J3)
whose closure is not orientable (here O4
and J3 are the branches of the
quartic and cubic which meet each other in 12 points).
Below each picture in Section 5.4, we refer to its realization(s).
The references are of the following types:
-
Reference to the figure with a singular curve which is to
be perturbed.
-
Reference to a paper where the curve is constructed.
-
2+3. This means that the arrangement is obtained from an
arrangement of a conic C2
= {f = 0} and a cubic C3
by replacing C2
with {f²+ tg = 0}
for some polynomial g of degree 4 and a small parameter
t.
-
2+4. This means that the arrangement is obtained from an
arrangement of a conic C2
= {f = 0} and a quartic C4
using the method introduced by Polotovskii in [11], i.e. by replacing
C2
with {f L + t g = 0} where t
is a small parameter,
g is some polynomial of degree 3, and
{L = 0} is a line chosen as it is explained in [11].
-
$x^y$ where x = 1, ..., 11 and y
= 1,2, ... The construction similar to that depicted in Figure 53
with the following initial data. The arrangement in Figure 52.x
with the point indicated by y as the point p.
Example: 2² is the construction
in Figure 53.
-
$x^y$ where x = 1, ..., 11 and y
= a,b, ... The construction similar to that depicted in Figure 54
with the following initial data. The arrangement in Figure 52.x
and the point p is obtained by a degeneration of the
digon indicated by y . Example: $8^c$ is the
construction in Fingure 54.
The arrangements in Section 5.4 are ordered lexicographically
according to the word w which encodes the mutual arrangement
of J3 and
O4. The word w
is defined as follows. Let us numerate the intersection points by the characters
1, 2, ..., 9,
a, b, c clockwise along the oval O4
starting from the leftmost point. The word w is
composed of these characters placed according to their order along
J3.
The symbol "/" means a "passage through the infinity".