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Oberseminar WiSe 12/13 Zusammenfassung des Vortrags von Stefano Pascolutti

Modular Forms and the Looijenga Conjecture.

An intriguing conjecture by E. Looijenga states that the moduli space M_g of smooth curves of genus g is the union of g-1 affine open subsets. In a joint work with C. Fontanari, we prove that the conjecture is true for 2 \leq g \leg 5 by giving an explicit open covering. In this talk, I will give a brief introduction on modular forms and theta functions, together with some remarkable examples and their geometrical/arithmetical meaning. I will make use of these examples to prove the conjecture for 2 \leq g \leq 5.