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Let $C$ be a smooth projective curve and $X=\mathbb{P}_C(E)$ be a ruled surface over $C$. Let $x_1,\ x_2\in X$ be closed points and define $X_1,\ X_2$ to be elementary transforms of $X$ at $x_1,\ x_2$ …

I am now trying to construct a $Z_2\times Z_2$-cover over $\mathbf{P}^n$. From the paper of Pardini, we need line bundles $L_1$, $L_2$, $L_3$ and divisors $D_1$, $D_2$, $D_3$ which satisfies the follo …

Fano varieties are defined by the ampleness of $-K_X$, and a rough statement of a step in the Mori program is to check whether a variety is a Fano-fibered one. By that reason, Fano ones are important …

It is hard to find any reference which contains a proof of the following statement: slope stability is an open condition in a flat family. There is one I have found, '65 paper of Narasimhan and Sesha …

The base field is algebraically closed and of chatacteristic zero. If $X$ is a smooth projective curve and $Y\to X$ is an étale covering of $X$ of degree $d$, then what can we say about the automorphi …

It is about deformation theory on algebraic surfaces. If there are two singular points on an algebraic surfaces, is it possible that two singular points collapse to a new point as the surface deforms? …