Thuses | Math discussions

The rank of $y^2 = x^3 - 2$ via Mazur-Tate methods

When I was a young kid, I heard the mathematical fact that the only (positive) integer that is one more than a square and one less than a cube is $26$ . Said differently, the only integer solutions $(x,y)$ to $y^2 = x^3 - 2$ are given by $(3,\pm 5)$ . There are elementary methods to prove this, using …

Isometries of a product of Riemannian manifolds

Theorem. Let $(A, g_A)$ and $(B, g_B)$ be two compact Riemannian manifolds with irreducible holonomy groups. Let $(X, g):= (A \times B, g_A \oplus g_B)$ . Then

$\operatorname{Isom}(X) = \begin{cases} \operatorname{Isom}(A) \times \operatorname{Isom}(B), \ \text{ if } A \text{ is not isometric to } B \\ \left ( \operatorname{Isom}(A) \times \operatorname{Isom}(B) \right ) \rtimes \mathbb{Z}/2\mathbb{Z}, \ \text{ otherwise } \end{cases}$

This result seems to be a folklore, probably well known to the specialists, although it is hard to find it in the literature. The only discussion which I managed to find on Mathoverflow contains …

Finite flat commutative group schemes embed locally into abelian schemes

Let $G$ be a finite flat commutative group scheme over a fixed locally noetherian base scheme $S$ . In this brief note, I want to explain the proof of the following theorem due to Raynaud.

Theorem. There exists, Zariski-locally on $S$ , an abelian scheme $A$ such that $G$ embeds as a closed $S$ -subgroup of …

What Topological Spaces are π₀?

This was a fun question I thought about once. My answer is at the end, in case you’d like to try solving the problem yourself. The question is likely more interesting than my solution.

A well known theorem says that every group occurs as $\pi_1(X)$ for some topological space $X$ . It’s …

A curiosity: “supersmooth” varieties

I want to share a curious condition on varieties for which I have found no use. Let $k$ be a field and let $X$ be a locally finite type $k$ -scheme. Recall that $X$ is said to be smooth if, for every Artin local $k$ -algebra $A$ with a proper ideal $I \subset A$ and every $k$ -morphism $\Spec A/I \to X$ , there …

A proof of a general slice-Bennequin inequality

In this blog post, I’ll provide a slick proof of a form of the slice-Bennequin inequality (as outlined by Kronheimer in a mathoverflow answer.) The main ingredient is the adjunction inequality for surfaces embedded in closed 4-manifolds. To obtain the slice-Bennequin inequality (which is a statement about surfaces embedded …

The Picard number of a Kummer K3 surface

Let $k$ be a separably closed field of characteristic not $2$ , and $A/k$ an abelian surface. Then it is a basic fact (e.g. see Example 1.3 (iii) of Huybrechts’ “K3 Surfaces”) that one can make a K3 surface out of $A$ . The construction is as follows. Consider the involution $\iota : A \to A$ given by $x \mapsto -x.$ The …

Brieskorn resolutions via algebraic spaces

I’d like to discuss simultaneous resolutions of surfaces from a moduli-theoretic perspective, following Michael Artin’s paper on Brieskorn resolutions.

Artin’s approach to moduli begins with the most desirable aspect of a moduli space, its universal property. That is to say, define a functor the space should represent and then check …

The torsion component of the Picard scheme

This post is a continuation of Sean Cotner’s most recent post [see An example of a non-reduced Picard scheme]. Since writing that post, Bogdan Zavyalov shared some notes of his proving the following strengthened version of the results described there.

Main Theorem. Let $S$ be a noetherian local ring and …