Introduction to Newton-Okounkov bodies
I will recall the construction of Newton-Okounkov bodies for pseudo-effective divisors on normal projective varieties and discuss a few natural related questions. I will then focus on the two dimensional case and present some recent results on the geometry of Newton-Okounkov bodies and global Okounkov bodies.
Functions associated to valuations on the Okounkov body
In this talk we will review recent work by Boucksom, Chen, Nystrom et al. encoding valuations as a convex function on the Okounkov body.
The Universal Minkowski summand of a polyhedron
Given a convex polyhedron Q, one can ask for the
possibilities of decomposing it into a sum Q = A + B
of simpler polyhedra. Some of these splittings can
be considered as extremal decompositions - but the
interesting point is that there might be several ones,
i.e. Minkowski prime decomposition is not unique.
This structure is captured in the polyhedral cone C(Q)
of Minkowski summands of Q. It comes with a so-called
universal Minkowski summand C'(Q) being a cone with a
projection onto C(Q). This construction generalizes the
well-known Cayley polytopes.
Via toric geometry, these objects can be brought into
algebraic geometry in two different ways: On the one hand
they lead to local and global Newton-Okounkov bodies;
but using the dual setup, they describe features in
deformation theory of toric varieties.
Generic vanishing theory and equivalences of derived categories
In the first part of the talk I will discuss and motivate a few
results in generic vanishing theory by giving particular emphasis to the study of linear series on irregular varieties. In the second part of the talk I will illustrate the behavior of non-vanishing loci, the main objects in generic vanishing theory, under equivalences of derived categories. Finally, time permitting, I will show an application of this behavior by studying fibrations onto smooth curves of genus at least two
under derived equivalences.
Some counterexamples on blow-ups of P3
I will discuss how the classical geometry of point blow-ups of \mathbb P3 can be used to construct some interesting examples in birational geometry: a nef divisor which is 0 on a countable set of curves, a quasi-projective variety with only a countable set of complete, positive-dimensional subvarieties; and an infinite set of non-isomorphic varieties that all have equivalent derived categories of coherent sheaves. Most of this is joint with John Christian Ottem.
On the effective cone of Pn blown-up at n+3 points
We start with an overview on Interpolation Problems. We study linear systems of hypersurfaces of a fixed degree passing through a collection of n+3 general points with assigned multiplicities. The rational normal curve of degree n passing through the points, its secant varieties and joins with linear subspaces are cycles of their base locus and we compute their multiplicity of containment. We give the facets of the effective and movable cones of divisors on blown-up projective spaces. This yields a conjectural formula for the dimension of such linear systems, completing a conjecture in the commutative algebra setting due to Froeberg-Iarrobino.
This is joint work with M. C. Brambilla and E. Postinghel
Infinitesimal Okounkov bodies on surfaces
Fix a big divisor D on a projective algebraic surface X, and a smooth point p on X. Generalizing the notion of infinitesimal Okounkov body (first studied by Lazarsfeld-Mustata , see also Küronya-Lozovanu arXiv:1411.6205) consider the family of all Okounkov bodies of D with respect to a flag (E,x) which is infinitely near to p, in the sense that there is a sequence of point blowups Y->X mapping the irreducible rational curve E to p.
Some questions that I will address are: which properties of D can be read off from the collection of infinitesimal bodies; parametrisation and continuity/semicontinuity of such bodies; relationship with Seshadri constants; etc.
This is work in progress, and I am completely open to studying these questions within a working group during the workshop if more people are interested.
Line arrangements with the maximal number of triple points
Configurations of lines since Pappus have been a classical object of
interest in geometry. Recently they played a prominent role in the
construction of counter-examples to the Containment Problem and in
some works related to the Bounded Negativity Conjecture. On the
other hand, configurations on their own have been studied in combinatorics for example by Bokowski and in algebraic combinatorics
for example by Sturmfelds. I will explain why configurations with
certain numerical invariants are of interest in algebraic geometry
and I will present some results obtained recently jointly with the
Bounded negativity and arrangements of elliptic curves.
The H-constant and its variant, the linear H-constant, have been introduced in order to have a better understanding of the behavior of the bounded negativity conjecture under blow-ups. In this talk I propose to study the elliptic H-constant and give particular examples of elliptic curves configurations in the plane implying that the H-constant of any surface is less or equal to -4.
Geography of simply connected surfaces of general type
The regularity of certain subschemes of the singular locus of hypersurfaces
We revisit Dimca˘s method to compute the Alexander polynomial of a hypersurface X in Pn with only isolated quasihomogeneous singularities.
We use Dimca˘s method to bound the regularity of several subschemes of the singular locus of X. As an application we prove the following result:
Let I be the ideal of finitely many points in Pn. Then for m ³ n we have that all forms f in I(m) of degree at most (m/n) reg S/I+m-1 define a hypersurface with nonisolated singularities.
Severi varieties and linear systems