Counting tropical rational curves with cross-ratio constraints

Abstract

This is a follow-up paper of [Gol18], where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in [Tyo17] allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension.In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio conditions. Moreover, graphical contributions are introduced which provide a novel and structured way of understanding multiplicities of floor decomposed curves in R 3 . Additionally, socalled condition flows on a tropical curve are used to reflect how conditions imposed on a tropical curve yield different types of edges. This concept is applicable in arbitrary dimension. Notation 1.1. We write [m] ∶= {1, . . . , m} if 0 ≠ m ∈ N, and if m = 0, then define [m] ∶= ∅.Underlined symbols indicate a set of symbols, e.g. n ⊂ [m] is a subset {1, . . . , m}. We may also use sets S of symbols as an index, e.g. p S , to refer to the set of all symbols p with indices taken from S, i.e. p S ∶= {p i i ∈ S}. The #-symbol is used to indicate the number of elements in a set, for example #[m] = m.

Generalizing Tropical Kontsevich's Formula to Multiple Cross-Ratios

“…Thus all our maps can be treated as maps to either R 2 or R 1 . Hence Proposition 1.15 of [Rau16] can be applied, and together with Proposition 1.12 of [Rau16] and Lemma 2.11 of [Gol20] it follows that the support of the intersection product…”

“…The general Kontsevich's formula we derive this way allows us to recover Kontsevich's fomula, see Corollary 71. The initial values of the general Kontsevich's formula are the numbers provided by the original Kontsevich's formula and so-called cross-ratio multiplicities, which are purely combinatorial [Gol20].…”

“…We recall some standard notations and definitions from tropical geometry [Mik07,GM08,GKM09] and give a very brief overview of the necessary tropical intersection theory [FS97,Rau09,All10,AR10,Kat12,Sha13,AHR16,Rau16]. After that, tropical cross-ratios and degenerated tropical cross-ratios are defined [Gol20].…”

“…As indicated in the last section, tropical intersection theory can be applied to the tropical moduli spaces that are interesting for us. For a short and -for our purposes -sufficient introduction to tropical intersection theory have a look at the preliminary section of [Gol20]. For more background of tropical intersection theory see [FS97,Rau09,All10,AR10,Kat12,Sha13,AHR16,Rau16].…”

“…Mikhalkin introduced so-called tropical double ratios in [Mik07] as a tropical analogue of classical cross-ratios. The author gave an intersection theoretic version of Mikhalkin's definition in [Gol20]:…”