In the last 2 lectures I will:

1. introduce (higher) secant varieties, (weakly) defective varieties and recall Terracini’s lemma;
2. I will talk about the famous theorem of Severi which classifies 1- defective surfaces;
3. I will talk about terracini’s extension to higher secant surfaces;
4. I will prove a bound on the degree of secant varieties;
5. I will classify surfaces for which the bound is achieved, showing how all this is related to castelnuovo–enriques’ theorem above.

Parts 4 & 5 above is recent joint work with F. Russo.