Introduction to real del Pezzo surfaces

Del Pezzo surfaces are algebraic surfaces playing an important role in the classification of algebraic projective surfaces up to birational transformations. Over C, a smooth del Pezzo surface of degree d is either isomorphic to P²C (d = 9) or to P¹C × P¹C (d = 8) or to the blow-up of P²C in 1 ≤ r ≤ 8 points in general position, where d = 9 − r. Before explaining the terminology, we will first see how algebraic projective varieties defined over a perfect field (e.g. R later on) naturally arise while studying some fibrations over P¹C, motivating the interest in the classification of smooth (rational) del Pezzo surfaces over a perfect field. We will then introduce the notions of real structures and real forms of complex projective surfaces, and we will see that there are exactly two non-equivalent rational real models of P¹C ×P¹C. Finally, we will present the classification of smooth real rational del Pezzo surfaces of degree 6, that were initially introduced over C to be the blow-up of P²C in three points in general position.