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I am very new to algebraic geometry. Currently reading about Hilbert and quot scheme. I want to know more about the structure and properties of Hilbert and quot schemes over curves. My question is the ...

Suppose $X$ is a smooth projective surface over $\mathbb{C}$ with irregularity $0$ $(q_1(X)=0)$. I want to understand the curves on the Hilbert scheme of $n$-points on $X$. By the work of Fogarty, we ...

Suppose $\operatorname{Nef}(X)$ is a rational polyhedron with extremal rays $\{F_i\}_i$. Now, consider the Hilbert scheme of $n$ points $X^{[n]}$ and the embedding $\operatorname{Nef}(X)\subset \...

We work over a field of characteristic $0$. Let $X\hookrightarrow\mathbb P^3$ be a geometrically integral hypersurface of degree $\delta$. It is well known that the Hilbert scheme of conics in $\...

I am considering the following setting: Let $(Y_d, X_{4d+2})$ be the pair of degree $d$ and index 2 Fano threefold $Y_d$ and degree $4d+2$ index 1 Fano threefold and both of them are Picard number 1. ...

Is there some standard technique or approach to determine when a (irreducible) subvariety of a rationally connected variety is again rationally connected? Any reference/text dealing with this kind of ...