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Since the Lazarsfeld-Mukai bundle $E = E_{C,A}$ fits into the exact sequence $$0 \to H^0(C,A)^\vee \otimes \mathcal{O}_S \to E \xrightarrow{\phi} K_C(-A) \to 0,$$ the first Chern class of $E$ is $[ …

Are there complex projective elliptic threefolds $f: X \to S$ with infinitely many rational sections satisfying the following properties? $S$ is a smooth surface of Picard rank $1$. $X$ is smooth wit …

Let us do the example where $\mathscr{M}$ is a proper Deligne-Mumford moduli stack of varieties. Note that if $\mathscr{M}$ is one-dimensional, then the Cohen-Macauleyness of $\mathscr{M}$ is equival …

I suppose that you are discussing Betti cohomology with coefficient in $\mathbb{Q}$. Using the long exact sequence $$\cdots \to H^{k}(X,\mathbb{Q}) = H^{k+2r}(Y,Y \backslash X,\mathbb{Q}) \to H^{k+2r …

This is not an answer but rather a lengthy comment. A necessary condition for the pencil to be isotrivial is that a smooth member in that pencil has a non-trivial automorphism: By blowing-up the base …

For a reference in English, you could take a look at this paper of Varouchas, which contains a definition of Kähler spaces and relatied concepts (e.g. Kähler morphisms) and some of their fundamental …

If $k$ is an algebraically closed field of characteristic $0$, then the map $$X \mapsto N(X) := \#(\text{connected components of }X)$$ defined for smooth projective varieties extends to a ring homom …

This is related to Mori's theorem through Grauert's ampleness criterion in Hartshorne's "Ample vector bundles" (Proposition 3.5). Let's assume that $M$ is projective and $\dim M \ge 2$. Let $\alpha : …

Apart from the reference given in the comment, you can also find a proof of the existence of the fiber product in Fischer's "Complex Analytic Geometry", Corollary 0.32. For direct products, a more st …

Given an affine algebraic variety $V$ such that $\Gamma(V,\mathcal{O}_V)$ is a UFD, its sheaf of ring can be determined easily since one can show that: $$\Gamma(D(f_1) \cup \cdots \cup D(f_n),\mathca …

The Jacobian matrix consists of coefficients of $t^3,t^2u,tu^2,u^3$ in the following $4$ partial derivatives $$\partial_p F(t,u,pt+ru,qt+su) = \partial_{y}F(t,u,pt+ru,qt+su)t,\\ \ldots\\ \partial_s F( …

Here is a simple proof using Theorem 1.6.1 in Lazarsfeld book, which is the following: Theorem (Demailly) If $H_1,\ldots,H_n$ are Kähler classes in a compact Kähler manifold of dimension $n$, then th …

Let $C$ be a smooth projective curve (say over $\mathbf{C}$ for simplicity). Given $L,M \in \mathrm{Pic}^0(C)[n]$, recall that the Weil pairing $e_n(L,M)$ is defined as follows. First of all, for an …

Edit notice: As Evgeny Shinder pointed out in the comment, it is unclear why we have $S = J^{-1}(\{0,1,\infty\})$ where $J : C \to \mathbf{P}^1$ is the $j$-invariant map. The problem is that $X \to S$ …

Let $X$ be a complex compact Kähler minimal surface with zero algebraic dimension and $H^2(X,\mathcal{O}_X) \ne 0$. We know that according to Enriques–Kodaira classification, $X$ is either a torus or …