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A smooth variety X of dimension n is said to satisfy the diagonal property if there exists a vector bundle ε of rank n on X × Xand a section s of ε such that the image ∆(X) of the diagonal embedding of X into X × Xis the zero scheme of s. A study of varieties satisfying the diagonal property was begun by Pragacz, Srinivas and Pati, in [8]. Even though there are many cases where the answer is affirmative, only in a few examples an explicit description of the vector bundle is known. After an exposition of toric varieties, we discuss this question in the particular case when X is a toric surface, in search for such examples. [ABSTRACT FROM AUTHOR]

These are lecture notes of a short course on the moduli stack of vector bundles on an algebraic curve. The aim of the course was to use this example to introduce the notion of algebraic stacks and to illustrate how one can work with these objects. Applications given are the (non-)existence of universal families on coarse moduli spaces and the computation of the cohomology of the moduli stack. [ABSTRACT FROM AUTHOR]