5 problems

Let $f({\boldsymbol x})\in {\mathbb Z}[x_1,x_2,\ldots,x_n]$ be a homogeneous polynomial of degree $d$, and let $V\subset {\mathbb P}^{n-1}$ be the projective variety defined by…

Let $K$ be a number field, let $\mathcal{S}$ be a finite set of places, and write $K_{\mathcal{S}}=\prod_{v\in\mathcal{S}}K_v$ and $\mathcal{O}$ for the ring of $\mathcal{S}$-integ…

Let $M_f$ be the Milnor algebra of a form $f$, let $X_n^d$ be the variety of degree-$d$ forms in $\mathcal Q_n^d$ with nonzero discriminant, and let … be the associated form morphi…

Let $K$ be a number field with a set of places $\mathcal{S}$, let $K_{\mathcal{S}}=\prod_{v\in\mathcal{S}}K_v$, and let $\mathcal{O}$ be the ring of $\mathcal{S}$-integers. Let…

Let $X_m^n$ be the space of homogeneous forms of degree $n$ in $m$ variables, with $n\geq 3$. For $Q\in X_m^n$, let ${\mathcal R}_m^n$ be the collection of invariant rational funct…