Victor Lozovanu

Local positivity of linear series on surfaces

We study asymptotic invariants of linear series on surfaces with the help of Newton-Okounkov polygons. Our primary aim is to understand local positivity of line bundles in terms of convex geometry. We work out characterizations of ample and nef line bundles in terms of their Newton-Okounkov bodies, treating the infinitesimal case as well. One of the main results is a description of moving Seshadri constants via infinitesimal Newton-Okounkov polygons. As an illustration of our ideas we reprove results of Ein-Lazarsfeld on Seshadri constants on surfaces.

Infinitesimal Newton–Okounkov bodies and jet separation

In this paper we explore the connection between asymptotic base loci and Newton-Okounkov bodies associated to infinitesimal flags. Analogously to the surface case, we obtain complete characterizations of augmented and restricted base loci. Interestingly enough, an integral part of the argument is a study of the relationship between certain simplices contained in Newton-Okoukov bodies and jet separation; our results also lead to a convex geometric description of moving Seshadri constants.

A few questions about curves on surfaces.

In this note we address the following kind of question: let X be a smooth, irreducible, projective surface and

Positivity of line bundles and Newton-Okounkov bodies

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Volume functions of linear series

D a divisor on X satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is effective or movable? We describe some examples, discuss some conjectures and prove some results that suggest that the answer should in general be negative, unless one puts some really strong hypotheses either on

A Reider-type theorem for higher syzygies on abelian surfaces

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