When I was a graduate student, Zev Rosengarten (a former student of Brian Conrad) and I used to eat dinner at Stanford’s Arrillaga dining hall a lot. We’d talk about math for hours, but one thing that will forever be ingrained in my mind is how Zev was able…

Tokuyama’s Formula is a combinatorial result that expresses the product of the deformed Weyl denominator and the Schur polynomial as a sum over strict Gelfand-Tsetlin patterns. This result implies Gelfand’s parametrization of the Schur polynomial, Weyl’s…

This post is doubling my old question asked on Mathoverflow. Let be a smooth projective complex variety of dimension , and let be an ample irreducible divisor. Recall that a cohomology class is called -primitive, if . Here, is the first Chern class of the…

The Willmore energy for a surface in Euclidean 3-space is defined as , where is the mean curvature of and its area form. It’s known to be invariant under the conformal transformations (whereas the mean curvature itself is not). White, and later Bryant noticed…

In the celebrated paper “Vector bundles over an elliptic curve” M. Atiyah classifies indecomposable vector bundles, namely he provides a bijection between indecomposable bundles of arbitrary rank and degree (denoted by ) and (where ). The latter is described…

Let be a smooth projective connected scheme over an algebraically closed field (experts will notice that several of these hypotheses can be weakened in what follows). Attached to is the Picard scheme , a locally finite type -scheme defined functorially as…

1.Preliminaries: Chern theorem This post is a result of several discussions with Rodiohn Déev. Fix a complex manifold and a complex vector bundle over . Recall that a structure of holomorphic bundle on is given by an operator     which satisfies -Leibniz…

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…

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.…

Let be a separably closed field of characteristic not , and 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 . The construction is as follows. Consider the involution…

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…

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

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 for some topological…

July 16, 2021. We introduce lazy LaTeX render. Now, when you open a page with new formulas, it uploads instantly and then formulas start rendering in front of your eyes. This way, we can avoid server overload and make pages open faster even if they have thousands…

Let be a finite flat commutative group scheme over a fixed locally noetherian base scheme . In this brief note, I want to explain the proof of the following theorem due to Raynaud. Theorem. There exists, Zariski-locally on , an abelian scheme such that embeds…

Theorem. Let and be two compact Riemannian manifolds with irreducible holonomy groups. Let . Then 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…

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 . Said differently, the only integer solutions to are given by . There are elementary methods to prove this…

In this post I want to give a few examples of the known “pathological” behavior of algebraic groups defined over general bases. In particular, this post contains examples of the following. A smooth group scheme over a DVR with generic fiber and special fiber…

When people say finite flat group scheme, what exactly do they mean? Sometimes, they just mean finite flat group scheme, presumably over some prefixed base. But often people meant finite flat *commutative* group scheme. It’s confusing. In this blog, I shall…