Mathematics Colloquia and Seminars

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Given a variety embedded in projective space, one can associate invariants, called Betti numbers, defined via the minimal free resolution of its homogeneous coordinate ring. A long standing question has been to understand what geometric information these invariants contain. In the case of a curve embedded via the canonical linear system, a theorem of Voisin tells us which of these Betti numbers vanish in terms of a geometric quantity called the gonality (up to a genericity assumption). We will consider the question of relating the values of the non-zero Betti numbers to the geometry of the curve. In particular, we prove that the last Betti number in the linear strand of the resolution tells us how many minimal pencils the curve possesses, modulo some genericity assumptions.