Let $G$ be an affine, connected, reductive group and $X$ a $G$-module. Choose a maximal torus $T\subseteq G$, a Borel $B\subseteq G$ containing $T$ and let $U$ be the unipotent radical of $B$. Denote by the character group of $T$. Let $\Lambda\subseteq\mathbb{X}$ be the set of dominant weights of $G$ with respect to these choices. We can decompose the graded coordinate ring $\mathbb{C}[X]=\bigoplus_{\lambda\in\Lambda} V_{(\lambda)}$ into its isotypic components $V_{(\lambda)}$ of weight $\lambda$. Let
\[ \Lambda_X=\{ \lambda\in\Lambda \mid V_{(\lambda)}\ne\{0\}\} \]
be the set of weights that occur in $\mathbb{C}[X]$.
Let $V_{(\lambda)}\cong V_\lambda^{\oplus n_\lambda}$, where $V_\lambda$ is the irreducible module of highest weight $\lambda$. Each $V_\lambda$ has a highest weight vector which is unique up to scaling - let $f_{\lambda 1}, \ldots, f_{\lambda n_\lambda}\in V_{(\lambda)}$ be linearly independent highest weight vectors.
Claim. *For each $1\le k\le n_\lambda$, if the function $f:=f_{\lambda k}\in\mathbb{C}[X]$ is reducible, then there exist weights $\lambda_1,\ldots,\lambda_r\in\Lambda_X$ such that $\lambda$ is an $\mathbb N$-linear combination of the $\lambda_i$.*
Indeed, about a year ago this statement was completely unclear to me. However, it's actually not that hard to see and I felt like sharing my proof. (more…)
I was talking to this fellow mathematician, and told him how for my very personal taste, the research in algebraic geometry has sailed a bit too far ashore from the original question that motivated the subject: A classification for the solutions of polynomial equations. I told him that most of the heavy machinery in algebraic geometry can be given very good intuition - however, most courses neglect to do so.
Anyways, he was clearly amused and not quite in belief of my radical position that unconditional and solitary abstraction isn't the only way to do algebraic geometry. He teased me to then tell him what an *»ample«* line bundle was, without talking about tensor powers or commutative diagrams.
So, that's the reason for this post. I answered that ample line bundles are precisely those that induce finite morphisms to projective space, but I couldn't remember where I knew it from. I searched through all the major literature and could not find a reference until someone on Mathoverflow helped me out. Since I complained so much about how noone ever gives intuition to these kinds of concepts in algebraic geometry, I decided I'd blog a bit about line bundles and how you should think about them.
Do you want to know more?