## T-time: one day meeting on toric methods in algebra and geometry

The meeting will be held on the 12th January 2017 in the Alan Turing Building at the University of Manchester.

All talks take place in Frank Adams Room 1.

Anyone interested is welcome to attend.
Some limited funds may be available to contribute to the expenses of research students who wish to attend the meeting.

For enquiries contact: hendrik.suess@manchester.ac.uk

The meeting is supported by an LMS conference grant.

For advertising the event: a poster

### Abstracts

#### Hendrik Süß: Semi-toric integrable systems and toric degenerations

Smooth toric varieties come naturally with the structure of an integrable system by considering the moment map of the torus action. In the case of lower dimensional torus actions the situation is more complicated. Here, a recent result by Hohloch, Sabatini, Symington and Sepe gives an exact criterion for the existence of so-called semi-toric systems on an algebraic surface with $\mathbb{C}^*$-action. We compare symplectic and algebraic results for such surfaces and conclude that the existence of semi-toric systems on $\mathbb{C}^*$-surfaces is equivalent to the existence of an equivariant degeneration to a normal toric variety. This is joint work with Christophe Wacheux.

#### Jonathan Fine: Linear homology in a nutshell

Every convex polytope X has a flag vector $f(X)$. Cartesian product on convex polytopes induces a ring structure on the span of all convex polytope flag vectors. The flag vector ring has operators cone, product and $D$ (difference between product with square and triangle).
We seek a basis for the flag vector ring, for which the structure constants for cone, product and D are all non-negative integers. When we write $f(X)$ in this basis, the coefficients are linear functions of $f(X)$. The speaker expects these coefficients to also be non-negative integers, being the Betti numbers for an as yet unknown homology theory $G$, which he calls linear homology.
Linear homology extends middle perversity intersection homology. It provides additional Betti numbers that detect singularities. By a result of Bayer and Billera, convex polytopes of dimension d would have $F_{d+1}$ independent linear homology Betti numbers (where $F_{d+1}$ are the Fibonacci numbers).

#### Milena Hering: Frobenius splittings of toric varieties

Varieties admitting Frobenius splittings exhibit very nice properties. For example, many nice properties of toric varieties can be deduced from the fact that they are Frobenius split.
Varieties admitting a diagonal splitting exhibit even nicer properties. In this talk I will give an overview over the consequences of the existence of such splittings and then discuss criteria for toric varieties to be diagonally split.

#### Tom Coates: t.b.a.

#### Alexander Kasprzyk: Classifying del Pezzo surfaces via mutation