Two kinds of real lines on real del Pezzo surfaces of degree 1

Combined Count of Real Rational Curves of Canonical Degree 2 on Real Del Pezzo Surfaces With

“…On the other hand, due to Proposition 3.4.5 in [7] (where it is written in a bit different notation, so that becomes and turns into ). Since furthermore , we conclude that The third identity in the statement is nothing but the sum of the first two.…”

“…This work is based on our previous paper [7]. So we start with recalling its setup and principal ingredients.…”

“…It is the -structure of Theorem 1.1.1 that opened a way to a signed count of real lines in [7] where we introduced two species of real lines distinguished by the values of . Namely, we called a real line hyperbolic (resp.…”

“… ), and attributed to it an integer weight , . As was shown in [7], the count of real lines with these weights gives the following fundamental relations: where denotes the set of real lines in X ; and where is any Bertini pair.…”

“…It may be worth mentioning that our initial motivation was to study quadric sections of a real quadric cone that are 6-tangent to a fixed real sextic curve and to elaborate for them a count which would have as strong invariance properties as the count of real 3-tangent hyperplane sections established in [7] as one of the main applications of the relation (1.1.2) (see below Subsection 5.1). It is from analysis of the underlying wall-crossing phenomena that we came to an idea to combine together and and developed the corresponding system of weights.…”