Eoin Mackall
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Elliptic curves and their principal homogeneous spaces: splitting Severi--Brauer varieties Open
We consider the question: which elliptic curves appear as the Jacobian of a smooth curve of genus one splitting a Severi--Brauer variety? We provide three new examples. First, we show that if $E$ is any elliptic curve over an algebraically…
Pro-representability of $K^M$-cohomology in weight 3 generalizing a result of Bloch Open
We generalize a result, on the pro-representability of Milnor $K$-cohomology groups at the identity, that's due to Bloch. In particular, we prove, for $X$ a smooth, proper, and geometrically connected variety defined over an algebraic fiel…
On the Chow groups of a biquaternionSeveri–Brauer variety Open
We provide an alternative proof that the Chow group of $1$-cycles on a Severi--Brauer variety associated to a biquaternion division algebra is torsion-free. There are three proofs of this result in the literature, all of which are due to K…
On the Chow groups of a biquaternion Severi--Brauer variety Open
We provide an alternative proof that the Chow group of $1$-cycles on a Severi--Brauer variety associated to a biquaternion division algebra is torsion-free. There are three proofs of this result in the literature, all of which are due to K…
The Period and Index of a Generic Geometrically Elliptic Normal Curve Open
We construct genus one curves on base extensions of generic Severi--Brauer varieties of a given index and period which are versal objects for families of geometrically elliptic normal curves. We also compute the periods and indices of thes…
$\mathbb{A}^1$-homotopy equivalences and a theorem of Whitehead Open
We prove analogs of Whitehead's theorem (from algebraic topology) for both\nthe Chow groups and for the Grothendieck group of coherent sheaves: a morphism\nbetween smooth projective varieties whose pushforward is an isomorphism on the\nCho…
On intersection theory, Severi-Brauer varieties, and the intersection theory of Severi-Brauer varieties Open
This thesis investigates the Chow ring, and neighboring functors, of a Severi-Brauer variety. The approach taken here heavily depends on the computation of lower K-groups of a Severi-Brauer variety.We construct a functor (for an arbitrary …
Six variations on a theme: almost planar graphs Open
A graph is apex if it can be made planar by deleting a vertex, that is, there exists [math] such that [math] is planar. We also define several related notions; a graph is edge apex if there exists [math] such that [math] is planar, and con…