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In Butler' paper "Normal generation of vector bundles over a curve" (J.d.g.,1994). Proposition 1.4 said that $$prop^+(M_E)\leq \max\left\{-2,\frac{-prop^+(E)}{prop^+(E)-g}\right\}$$ where $E$ is a g …

The following reference may be helpful to you: [1] Yukio Matsumoto, José María Montesinos-Amilibia, Pseudo-periodic homeomorphisms and degeneration of Riemann Surfaces, Bull. Amer. Math. Soc., 30(19 …

Let $X,Y$ be smooth algebraic surfaces and $\pi:X\to Y$ be a double cover. Let $B\subseteq Y$ be the branch locus. We assume that $B$ is nef, big and smooth. [1] says that $\pi_1(X)\cong\pi_1(Y)$ (Se …

(1) Let $C\subseteq \mathbb{P}^2$ be a curve defined by a homogeneous equation $F(X,Y,Z)=0$. The dual curve of C, denoted by $C'$, is the image of the map $$[X,Y,Z]\to [\frac{\partial F}{\partial X},\ …

In Voisin's book "Hodge theory and complex algebraic geometry I", the proof of proposition 12.7 (page 296) says that if $X$ is projective, then every divisor $Z$ homologous to $0$ can be written as …

As I know, Sheng-li Tan is an expert on the linear systems on algebraic surfaces. You can refer to his papers such as [1]. I think you will find what you desire. In the case for ruled surface, you c …

In Voinsin's book [1], Theorem 11.32 (page 280) says: "If X is an algebraic variety, these subgroups of $Hdg^{2k}(X) coincide." However, the proof did not show that the subgroup generated by cycle …

Let $f:S\to S$ be a given periodic map of order $n$, and $(m_i,\lambda_i, \sigma_i)$ be the valency of the multiple point $p_i$ of $f$ ( $i=1,\cdots,r$ ). A classical result says such $f$ is isotop …

What is Mordell-Weil lattice?

Let $\Sigma_g$ be a Riemann surface of genus g, and $C$ is a cut curve of $\Sigma_g$, i.e. an oriented simple close curve. What is a right-handed Dehn twist of $C$ of $\Sigma_g$?

Let $f:X\to B$ be a family of curves of genus $g$ over a smooth curve $B$. Let $F_0$ be a singular fiber. If $F_0$ is a semistable fiber, the monodromy matrix can be gotten by the classical Picard-L …