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Consider the map $f:W \to Gr(2,n+1)$ taking a point $(x,y) \in W$ to the plane generated by $x$ and $y$y. Then the bundle under the question is the pullback of the tautological bundle on the Grassmann …

Such function never exists. Indeed, let $k$ be the order of $\Gamma$, then the degree of the branch divisor should be $km$ for some $m \ge 1$. Then $$ p_*O_\Sigma \cong O \oplus \psi \otimes O(-m) \op …

Let me give an answer for Hilbert schemes. Q. 1. Yes, the reason is that the functor $F_{X_1}$ is a closed subfunctor of the functor $F_X$ with respect to the natural embedding. In other words, for e …

This is true. The condition defining $F^X$ in $X\times Hilb(X)$ is the vanishing of the morphism $I_{Z,X} \subset O_X \to O_x$ (where $Z$ is a subscheme and $x$ is a point). It follows that the fiber …

The projection from a line $L_0$ is a birational isomorphism of $X$ onto $P^3$. It decomposes as the blow-up of the line $L_0$ followed by the contraction of a surface swept by lines intersecting $L_0 …

Some of them do, and some don't. Indeed, by global Torelli theorem, there is a K3 surface $X$ with $\mathrm{Pic}(X) = 0$. Such $X$ has no curves, in particular no rational curves. On the other hand, t …

The strict transform of $W$ in the blowup $\tilde{Q}_3$ of $Q_3$ at $O$ is equal to the blowup of $W$ at $O$, and since $W$ has a node at $O$, this blowup is the standard nonsingular model. To const …