Chennai Mathematical Institute


Wednesday, 21 July 2021, 3.30 pm
Bounded Negativity and Harbourne Constants on Algebraic Surfaces

Aditya N K Subramaniam
Chennai Mathematical Institute.


Let X be a nonsingular projective surface over an algebraically closed field. Bounded Negativity Conjecture (BNC) predicts that there is an integer b(X), depending only on X, such that the self-intersection C^2 is at least b(X) for every reduced curve C on X. J. Kollar gave a counterexample to this conjecture in positive characteristic. In characteristic zero, this conjecture is open in general though some easy cases are known. For example, if the anti-canonical divisor of the surface X is effective, the adjunction formula shows that X satisfies BNC. The conjecture is open in many interesting cases including when X is a blow up of the projective plane at finitely many points.

Harbourne constants were defined in an attempt to tackle this problem. Let C be a reduced curve on a smooth projective surface X. Harbourne constant H(X,C) of C is defined using the weighted self-intersection of the strict transforms of C on blow ups of X at finite sets of points. Harbourne constant of X is then defined as the infimum of H(X,C) over all reduced curves C on X. Harbourne constants allow us to study BNC on blow ups of X. For example, if the Harbourne constant of a surface X is finite, then BNC holds for all blow ups of X. There has been a lot of work on computing or bounding Harbourne constants in various situations. For example, Harbourne constants were studied for curve arrangements in the projective plane and in surfaces of non-negative Kodaira dimension.

Let X be a geometrically ruled surface over a smooth projective curve. In this thesis, we give lower bounds for the Harbourne constants of transversal arrangements of curves on X under some mild assumptions. We also define a global Harbourne constant as the infimum of Harbourne constants for arrangements of a specific type and give a lower bound for it. Finally, we show that the surfaces associated to transversal arrangements on ruled surfaces that we consider in this thesis are not ball quotients.