On $L$ -series of normal varieties

Algebraic Cocycles on Normal, Quasi-Projective Varieties

Blaine Lawson and the author introduced algebraic cocycles on complex algebraic varieties in [FL-1] and established a duality theorem relating spaces of algebraic cocycles and spaces of algebraic cycles in [FL-2]. This theorem has non-trivial (and perhaps surprising) applications in several contexts. In particular, duality enables computations of “algebraic mapping spaces” consisting of algebra...

The ‘ Varieties of Quotation ’ Series

Irrationality of Motivic Series of Chow Varieties

The Euler characteristic of all the Chow varieties, of a fixed projective variety, can be collected in a formal power series called the Euler-Chow series. This series coincides with the Hilbert series when the Picard group is a finite generated free abelian group. It is an interesting open problem to find for which varieties this series is rational. A few cases have been computed, and it is sus...

Normal Varieties and Birational Correspondences

Integral Dependence and Normal Varieties

b + an−1b + · · ·+ a0 = 0. Example 1.2. Let A = Z and B = Q. Then b ∈ Q is integral over Z if and only if b ∈ Z. Indeed, if b ∈ Z then b solves x − b ∈ Z[x]. Conversely, write b = c/d for relatively prime integers c and d, d > 0. Let f(x) = x + an−1x + · · ·+ a0 be a polynomial with integer coefficients that b satisfies. Substituting b for x and multiplying by d we obtain c = −(dan−1c + · · ·+ ...