**Title:** Roth's theorem for arbitrary varieties

**Speaker:** Michael Roth

**Speaker Info:** Queen's University

**Brief Description:**

**Special Note**:

**Abstract:**

If X is a variety of general type defined over a number field $k$, then the Bombieri-Lang conjecture predicts that the k-rational points of X are not Zariski dense. The conjecture is a prediction that a global condition on the canonical bundle (that it is ''generically positive'') implies a global condition about rational points. By the local-global philosophy in geometry we should look for local influence of positivity on the accumulation of rational points. To do that we need measures of both these local phenomena.Let L be an ample line bundle on X, and x \in X(\overline{k}). The central theme of the talk is the interrelations between $\alpha_x(L)$, an invariant measuring the accumulation of rational points around x as gauged by L, and the Seshadri constant \epsilon_{x}(L), measuring the local positivity of L near x. In particular, the classic approximation theorem of K.F. Roth on \mathbf{P}^1 generalizes as an inequality between \alpha_{x} and \epsilon_{x} valid for all projective varieties.

This is joint work with David McKinnon.

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