Unbounded negativity on rational surfaces in positive characteristic

We give explicit blowups of the projective plane in positive characteristic that contain smooth rational curves of arbitrarily negative self-intersection, showing that the Bounded Negativity Conjecture fails even for rational surfaces in positive characteristic. As a consequence, we show that any surface in positive characteristic admits a birational model failing the Bounded Negativity Conjecture.

Automorphism Groups of Certain Enriques Surfaces

We calculate the automorphism group of certain Enriques surfaces. The Enriques surfaces that we investigate include very general n-nodal Enriques surfaces and very general cuspidal Enriques surfaces. We also describe the action of the automorphism group on the set of smooth rational curves and on the set of elliptic fibrations.

When is M 0,n (ℙ1,1) a Mori dream space?

The moduli space M ¯ 0, n ( ℙ 1 , 1 ) ${{\bar{M}}_{0,n}}\left( {{\mathbb{P}}^{1}},1 \right)$ of n-pointed stable maps is a Mori dream space whenever the moduli space M ¯ 0 , n + 3 of ( n + 3 ) ${{\bar{M}}_{0,n+3}}\; \text{of} \;(n+3)$ pointed rational curves is, and M ¯ 0 , n ( ℙ 1 , 1 ) ${{\bar{M}}_{0,n}}\left( {{\mathbb{P}}^{1}},1 \right)$ is a log Fano variety for n ≤ 5.

Counting tropical rational space curves with cross-ratio constraints

This is a follow-up paper of Goldner (Math Z 297(1–2):133–174, 2021), where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in Tyomkin (Adv Math 305:1356–1383, 2017) 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. The multiplicities of such cross-ratio floor diagrams can be calculated by enumerating certain rational tropical curves in the plane.

Twelve Rational curves on Enriques surfaces

Given $$d\in {\mathbb {N}}$$ d ∈ N , we prove that any polarized Enriques surface (over any field k of characteristic $$p \ne 2$$ p ≠ 2 or with a smooth K3 cover) of degree greater than $$12d^2$$ 12 d 2 contains at most 12 rational curves of degree at most d. For $$d>2$$ d > 2 , we construct examples of Enriques surfaces of high degree that contain exactly 12 rational degree-d curves.

Quantum K-Theory of Grassmannians and Non-Abelian Localization

In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the q-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including level structures, as well as the Jackson-type integrals playing the role of equivariant K-theoretic mirrors.