On enumerative problems for maps and quasimaps: freckles and scars

Abstract

We address the question of counting maps between projective spaces such that images of cycles on the source intersect cycles on the target. In this paper we do it by embedding maps into quasimaps that form a projective space of their own. When a quasimap is not a map, it contains freckles (studied earlier) and/or scars, appearing when the complex dimension of the source is greater than one. We consider a lot of examples showing that freckle/scar calculus (using excess intersection theory) works. We also propose the "smooth conjecture" that may lead to computation of the number of maps by an integral over the space of quasimaps.

Figures (22)

• Figure 1: Configuration of source and target cycles in the $0+2 = 2$ example. We have two fixed points $c_1^X = (0:1) = 0$, $c_2^X = (1:0) =\infty$ in the source and one running point $x$ (the circle around $x$ indicates it is running.) Number of lines $\lambda_i$ through a source point denotes the codimension of corresponding target cycle $c_i^Y$ which is given as intersection of $\lambda_i$ hyperplanes in the target $\mathbb{P}^2$ (in this case, $\lambda_i \equiv 2$.)
• Figure 2: Configuration of source and target cycles in the $1+1 = 2$ example. We have three fixed points $c_1^X = (0:1) = 0$, $c_2^X = (1:1) =1, c_3^X = (0:1) = \infty$ in the source and one running point $x$. Target cycles are hyperplanes $c_1^Y, c_2^Y$ and points $c_3^Y,c_4^Y$.
• Figure 3: $4=1+3$ enumerative problem
• Figure 4: Quasiplanes in $\mathbb{P}^3$ passing through three points and two lines.
• Figure 5: One of the two 1-freckle configurations

Theorems & Definitions (53)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Proposition 2.6
• Remark 2.7
• Remark 2.8
• Example 2.9
• Example 2.10