## Geometry Seminar - Abstracts

### Talk

Thursday 14 December 2023, 10:30 - 11:30 in HG03.085

**Julian Demeio** (Bath)

*Weak weak approximation on Del Pezzo surfaces of low degree*

### Abstract

Swinnerton-Dyer showed in 2001 a weak approximation result for cubic surfaces.
For example, he proves that, if \(N\) is a co-prime number with \(6\), any mod \(N\)
solution of \(2 X_0^3=X_1^3+X_2^3+X_3^3\) lifts to an integral solution. Cubic
surfaces can also be defined as so-called "del Pezzo" surfaces of degree 3. In
the work I am going to present we extend the Swinnerton-Dyer result to del
Pezzo surfaces of degree 2. For example, we demonstrate that there exists an
integer S for which, for every integer \(N\) coprime with \(S\), any mod \(N\) solution of
\(2 X_0^2 = X_1^4+X_2^4+X_3^4\) lifts to an integral solution. Work in
collaboration with Sam Streeter and Rosa Winter.

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