Every rational surface is birational to a conic bundle or a del Pezzo surface. In this talk, we consider surfaces over finite fields that are both del Pezzo surfaces and conic bundles. We focus on surfaces which are minimal and have degree (defined as the self-intersection number of the canonical divisor) equal to \(1\). We give a classification based on the configuration of the singular fibres of the conic bundle structure, and discuss an answer to the following question: for which values of q can each of these configurations exist on a del Pezzo surface with conic bundle?

One approach to study a singularity \((X,0) \subseteq (\mathbb{C}^N, 0)\) is deformation theory. The central idea is to deform \((X,0)\) slightly into "simpler" singularities \((\mathcal{X}_t,0)\), while retaining some geometric properties of \((X,0)\). Naturally one can ask which singularities can only be deformed into finitely many different other singularities up to isomorphism, these are the so-called simple singularities. In the previously well studied classes of hypersurface and complete intersection singularities all simple singularities are isolated. For the class of Cohen-Macaulay codimension \(2\) (CMC\(2\)) singularities the existence of non-isolated rigid, in particular simple, singularities has been known since 1974. In this talk we will give an overview of the ongoing progress towards a complete classification of simple (non-isolated) CMC\(2\) singularities in recent years and discuss the difficulties of the still open subcase.

An elliptic surface \(Y\) over a smooth projective curve \(C\) is a smooth and projective variety \(Y\) of dimension 2 together with a fibration \(f: Y \rightarrow C\) such that the generic fibre is a smooth genus one curve. If \(Y\) is rational then there exists an \(m \geq 1\), called the index of the fibration, such that f is given by the pluri-anticanonical linear system \(|-mK_Y|\). The case \(m=1\) is equivalent to the existence of a global section for f. If \(m>1\), then f has a unique multiple fibre of multiplicity \(m\). A Halphen pencil of degree \(m\) is an irreducible pencil of plane curves of degree \(3m\) with nine possibly infinitely near base points of multiplicity \(m\). Rational elliptic surfaces of index \(m\) admit a realisation as a 9-fold blow up of the base points of Halphen pencils of degree \(m\). This allows for an explicit study of their singular fibres via the description of the special members of the corresponding pencils. In my thesis I described all possible constructions of rational elliptic surfaces of index 3 with fibres of type II*, III* or IV* by investigating the corresponding Halphen pencils.

Let \(\{X_t\}_t\) be a suitable family of germs at \(0\) of complete intersection varieties in \({\mathbb C}^n\) and \(\{f_t\}_t\), \(\{g_t\}_t\) families of non-constant polynomial functions on \(X_t\). If the germs \(X_t\), \(X_t \cap f_t^{-1}(0)\) and \(X_t \cap f_t^{-1}(0) \cap g_t^{-1}(0)\) are non-degenerate, locally tame, complete intersection varieties, for each \(t\), we prove that the difference of the Brasselet numbers \(B_{f_t,X_t}(0)\) and \(B_{f_t,X_t \cap g_t^{-1}(0)}(0)\) is related with the number of Morse critical points on the regular part of the Milnor fibre of \(f_t\) appearing in a morsefication of \(g_t\), even in the case where \(g_t\) has a critical locus with arbitrary dimension. This result connects topological and geometric properties and allows us to determine some interesting formulae, mainly in terms of the combinatorial information from Newton polyhedra.

Lunch is held at the canteen Uhlhornsweg. The menu can be found here.

We discuss a link, originally found by Gauss and Kroencker, between the number of integral points on the ellipsoid \(X^2+Y^2+Z^2=n\) and Hurwitz class numbers, and its extension to ellipsoids \(aX^2+bY^2+cZ^2=n\) for specific choices of integers \(a, b, c \in \mathbb Z\).

After the proof of Fermat's Last Theorem, the modular method has been used much more broadly to study solutions of Diophantine equations. In this talk, as motivation, we first briefly discuss solutions to the equation \(Ax^p+By^p=Cz^2\) over totally real fields satisfying certain local conditions. We then restrict to the equation \(x^p+dy^p=z^2\) over real quadratic fields \(\mathbb{Q}(\sqrt{d})\), and show that there exist explicit bounds (depending on \(d\)) such that no non-trivial solutions of a certain type exist whenever $p$ exceeds these bounds. In the end, we also describe what happens in the imaginary quadratic field setting. This is joint work with Erman Isik, Yasemin Kara, and Ekin Özman.