This question is all about paragraph $1$ of Sarkisov's article On Conic bundle structures. Here is a capture of the part I am interested in:
Let $k$ be an algebraically closed field whose characteristic is not $2$. A regular conic is a flat map of nonsingular (projective) varieties $\pi:V\rightarrow S$ whose generic fiber is an irreducible rational curve.
Consider a regular conic $(V,S,\pi)$. I have several questions about the first part of paragraph $1.5$:
- the author claims that $R^i\pi_*(\mathscr{O}_V(-K_V))=0$ for $i\geq1$. I can show that for $i>1$ (this is for example corollary $1.2$ in Hartshorne's Algebraic Geometry), but I don't know how to prove the case $i=1$.
- The author seems to deduce from the latter that $\mathscr{E}=(R^0\pi_*(\mathscr{O}_V(-K_V)))$ is a locally free sheaf of rank three and that $\mathscr{O}_V(-K_V)$ is relatively very ample and defines an embedding of $V$ in $Proj_S(\mathscr{E})$ under which each fiber of $\pi$ is a conic in $\mathbf{P}^2_k$. Why are all these assumptions true?