4 problems

Let $(f:X\rightarrow S)\in \mathcal{F}^n$ be a projective flat morphism of pure relative dimension $n$, and let $E$ be a vector bundle on $X$. Let $\lambda(E)$ denote the determina…

Let $(f:X\rightarrow S)\in \mathcal{F}^n$ be a projective flat morphism of pure relative dimension $n$ between schemes, and let $P$ be a homogeneous polynomial of degree $n+1$ in t…

Let $(f:X\rightarrow S)\in \mathcal{F}^n$ be a projective flat morphism of pure relative dimension $n$ between schemes, and let $L_0,\dots,L_n$ be line bundles on $X$. Write…

Let $\mathcal M$ be the moduli space of $n$-dimensional stable varieties with a fixed Hilbert polynomial. Using a universal stable family $\mathcal X$ over a finite cover of each i…