By James D. Lewis
This e-book presents an creation to a subject matter of critical curiosity in transcendental algebraic geometry: the Hodge conjecture. including 15 lectures plus addenda and appendices, the amount relies on a chain of lectures introduced through Professor Lewis on the Centre de Recherches Mathematiques (CRM). The booklet is a self-contained presentation, thoroughly dedicated to the Hodge conjecture and similar subject matters. It contains many examples, and so much effects are thoroughly confirmed or sketched. the inducement at the back of the various effects and heritage fabric is equipped. This entire method of the ebook offers it a ``user-friendly'' sort. Readers needn't seek in other places for varied effects. The ebook is acceptable to be used as a textual content for a issues direction in algebraic geometry; comprises an appendix by means of B. Brent Gordon.
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Extra info for A survey of the Hodge conjecture
By symmetry, C · D = (m + n − 4)e − (2h − 2), from which the formula as stated is immediate. 8 The same result can also be obtained by showing that the monodromy group E of 6 the family of smooth cubic surfaces acts transitively on the 432 pairs (E, L) as above on a ﬁxed S. D. Extrinsic pathologies 23 C is residual in this intersection to a quartic B of the form 2L + D with L a line and D a conic meeting L once. Show that L + D is the specialization of a twisted cubic disjoint from L and hence that C is a specialization of the generic element of J3 .
Since NS(S) is discrete, the class of C in it must be constant on any component of R. We therefore conclude that B’s of this type give rise to component(s) of R distinct from J3 . 8 We will simply list the steps, leaving the veriﬁcations as an exercise. First, the set of all pairs (L, E) is irreducible since the locus of E’s and L’s are PGL(4)-orbits in their respective Hilbert schemes. Second, over a dense open set in this base, the set of triples (S, L, E) such that S is a cubic surface containing L ∪ E forms a projective bundle, hence is again irreducible.
There is, however, an analogue Tg,n of Tg parameterizing marked surfaces of genus g with n marked points (or to use the indigenous terminology, with n punctures) from which 44 2. Basic facts about moduli spaces of curves been fruitfully exploited by Harer (whose results we’ll describe later) and, more recently, by Kontsevich. We should also mention that this approach provides Mg with a natural metric, called the Weil-Petersson metric whose positivity properties have been used by Wolpert (, ) to construct an embedding of Mg in a projective variety with many of the nice properties of the Deligne-Mumford stable compactiﬁcation which we will introduce later in this section.
A survey of the Hodge conjecture by James D. Lewis