PUBLIC VIVAVOCE NOTIFICATION Wednesday, 21 July 2021, 3.30 pm Bounded Negativity and Harbourne Constants on Algebraic Surfaces Aditya N K Subramaniam Chennai Mathematical Institute. 210721 Abstract 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 selfintersection 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 anticanonical 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 selfintersection 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 nonnegative 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.
