arXiv subjects
Publications and source records attributed to Sagar Kolte.
We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under suitable assumptions. We also provide examples which demonstrate sharpness of the results. The ambient surface is singular in these examples.
A pair $(S,C)$ is called a singular $\mathbb{Q}$-homology plane pair if $S$ is a singular projective surface with only quotient singularities having the same rational homology as $\mathbb{p}^2$ and $C \subset S$ has the same rational homology as $\mathbb{p}^1$. We will prove results concerning smooth rational curves on $S$ and the singularities of $S$ such that $\overline κ(S^0)=1$ and $\overline κ(S-C) \neq -\infty$. We end with an example of such pairs.
We will prove that given a genus-2 fibration $f: X \rightarrow C$ on a smooth projective surface $X$ such that $b_1(X)=b_1(C)+2$, the fundamental group of $X$ is almost isomorphic to $π_1(C) \times π_1(E)$, where $E$ is an elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic convexity of the universal cover of surfaces $X$ with genus-2 fibration $X\rightarrow C$ such that $b_1(X)>b_1(C)$.