| For a given arrangement of curves, we prove the existence of smooth projective surfaces with Chern numbers ratio arbitrarily close to the log Chern numbers ratio defined by the arrangement. This shows a strong connection between geography and log geography of algebraic surfaces. For instance, we apply this connection to produce several new examples of simply connected smooth projective surfaces with high Chern numbers ratio. The proof involves the use of a large scale behavior of Dedekind sums and continued fractions. In order for this to work, we need to consider random partitions of large primes numbers. Through examples, we show that randomness is indeed necessary by using a computer program. We also study arrangements of curves, their realizations, and conditions to exist. Using marked genus zero moduli spaces, we prove a one-to-one correspondence between certain arrangements and curves in projective space. In the case of lines, this correspondence allow us to find and classify (3, q)-nets for 2 ≤ q ≤ 6, and the Quaternion nets. |
