## Topics in the Geometry of Projective Space: Recent Work of F.L. ZakThe main topics discussed at the D. M. V. Seminar were the connectedness theorems of Fulton and Hansen, linear normality and subvarieties of small codimension in projective spaces. They are closely related; thus the connectedness theorem can be used to prove the inequality-part of Hartshorne's conjecture on linear normality, whereas Deligne's generalisation of the connectedness theorem leads to a refinement of Barth's results on the topology of varieties with small codimension in a projective space. The material concerning the connectedness theorem itself (including the highly surprising application to tamely ramified coverings of the projective plane) can be found in the paper by Fulton and the first author: W. Fulton, R. Lazarsfeld, Connectivity and its applications in algebraic geometry, Lecture Notes in Math. 862, p. 26-92 (Springer 1981). It was never intended to be written out in these notes. As to linear normality, the situation is different. The main point was an exposition of Zak's work, for most of which there is no reference but his letters. Thus it is appropriate to take an extended version of the content of the lectures as the central part of these notes. |

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### Contents

8 | |

Quadrics on a Severi variety | 19 |

Dimensions of Severi varieties | 26 |

The classification of Severi varieties | 33 |

41 | |

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### Common terms and phrases

2-plane action algebra applied argument assertion assume bundle choose classification clear Clearly codimension coincides complete intersection cone conjecture consider consists construct contains corresponding defined denote described dimension n/2 embedding equations exactly examples exists fact Figure follows four Furthermore given gives hand Hartshorne varieties Hence highest weight vector holds interesting irreducible isomorphism join least Lemma lies linear space linear system maximal meeting normal Note observe obtained orbit P'e Sec(X parametrized particular passing plane positive possible projective space projective variety proof Proposition prove quadratic quadric of dimension rank rational reference Remark representation respect secant quadric Qp secant variety Severi variety simple singular situation smooth quadric Span spinor variety stabilizer standard subgroup subspace subvarieties tangent theorem verify Zak's