Let \(X\) be a connected, projective curve over a perfect field \(\mathbb{K}\), and \(X_{\operatorname{red}}\) be the corresponding reduced curve. Then the natural closed immersion \(X_{\operatorname{red}} \to X\) induced an epimorphism \(\phi : \operatorname{Pic}(X) \to \operatorname{Pic} (X_{\operatorname{red}})\) between the Picard groups. We use this epimorphism to develop explicit methods for \(\operatorname{Pic}(X)\), if \(\operatorname{Pic(X_{\operatorname{red}}})\) is finitely generated. I give an overview of all the methods I use to get an explicit isomorphism \( \operatorname{Pic}(X) \cong \mathbb{K}^m \times \mathbb{Z}^r / U\), where \(U\) is a free \( \mathbb{Z}\)-submodule of \(\mathbb{K}^m \times \mathbb{Z}^r\) of finite rank. I then give a short demonstration of my implementation of this methods.

Specialization theorems by Néron and Silverman imply that in a family of Abelian varieties the Mordell-Weil rank of almost all members is at least the Mordell-Weil rank of the generic fiber. We study Mordell-Weil rank-jumps, that is, the locus of Abelian varieties in the family whose rank is strictly larger than the generic rank. In joint work with Cecília Salgado we show that in a family of Jacobians coming from a pencil of hyperellitic curves on a K3 surface the rank-jump locus is dense/not thin after a field extension of bounded degree.  

I give an overview over my master thesis. The starting point is a paper by Couveignes and Gasnier. They consider an unramified abelian galois cover \(Y \to X\) of two curves \(X\) and \(Y\) over a finite field with galois group \(G\), and study the effect of a \(G\)-action on two typical linear spaces on \(Y\) associated with pullbacks of divisors on \(X\). As one application they use this \(G\)-action to obtain better complexities for encoding and decoding algebraic-geometric codes. I describe extensions of this to ramified covers. 

Locally Recoverable Codes (LRCs for short) are a type of error-correcting codes particularly suited for applications in distributed storage systems. In this talk, I will present a construction of LRCs with availability from a family of fibered surfaces. Techniques coming from the theory of one-variable function fields and from the theory of fibrations on surfaces are combined to obtain the locality and availability properties of these codes, and to estimate their minimum distance. In particular, when the locality parameter is equal to 3, the bound that we obtain on the minimum distance is sharp.  

The field of post-quantum cryptography studies alternatives to currently existing cryptographic methods, as current methods are unsafe if a suitably powerful quantum computer is constructed. One approach is based on isogenies between elliptic curves, known as isogeny-based cryptography. While there are many isogeny-based protocols, until 2021 there was no isogeny-based commitment scheme. A commitment scheme allows a sender to show to a verifier that they committed to a certain message without revealing any information about the message. Moreover, in 2024 a modification to the isogeny-based commitment scheme was proposed that removes the need for a trusted third party during the setup phase. We discuss both of these protocols and study the theory behind them, which allows us to slightly generalise the proposed modification.

The plan is to have dinner at Finca & Bar Celona, Wechloy. The menu can be found here.