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Let $X\subseteq\mathbb{P}^n(\mathbf{C})$ be a complete intersection (smooth if you want). Q: Is there a good reference which gives (and proves in enough details) an explicit description of the graded …

So let $R$ be a discrete valuation ring and let $X$ be a scheme which is proper and flat over $R$. Let $X_s$ denote the special fiber of $X$. So intuitively, when somebody says that a curve $X$ is s …

Let $X$ be a smooth quasi-projective variety (so irreducible) over $\mathbf{C}$. We may think of $X$ as a complex manifold which we denote by $X^{an}$. Of course the topology on $X^{an}$ is finer tha …

Let $H_{\mathbf{Q}}$ and $H_{\mathbf{Q}}'$ be two pure Hodge structures of weight $n$ and $n'$ respectively. How do you prove the following simple fact: fact: If $n>n'$ and $f:H_{\mathbf{Q}}\rightarr …

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a p …

In Weil's short note entitled "On the Riemann hypothesis in function-fields" he mentions the notion of the complementary correspondence associated to a given correspondence $T:C\rightarrow C$ where $C …

I'm using the same setup as Corollary 1.7 on p. 44 of de Shalit manuscript (Iwasawa theory of elliptic curves with complex multiplication). I think there is a mistake in his Corollary 1.7 and I'm wo …

The proof of Corollary 1.7 is fine. I had misunderstood his proof. His proof uses in a crucial way his assumption (ii) which appears on the top of p. 41. As is explained on p. 41, this assumption impl …

Let $A$ be an abelian variety defined over $\mathbf{C}$ (of dimension $>1$) and let $\Theta_A$ be the holomorphic tangent sheaf of $A$. Question. How does one compute $H^1(A,\Theta_A)$ ? If $A$ …

Let $n\geq 2$ and $\mathbb{P}^n(\mathbf{C})$ be the complexe projective space of dimension $n$. Let $H\subseteq \mathbb{P}^n(\mathbf{C})$ be a hypersurface of degree $d$ where the coordinates in $\ma …

Let $\pi:X\rightarrow C$ be a flat and proper morphism over $\mathbb{C}$ where $X$ is a smooth projective surface and $C$ is a smooth projective curve. Assume that all the fibers of $\pi$, except fini …

Let $X\subseteq\mathbb{R}^k$ be a non-closed, unbounded semi-algebraic subset. Then it seems to me that Proposition 5.49 on p. 189 of the book Algorithms in Real Algebraic Geometry still holds true fo …

This is just an after thought about my question. At the end it is really a linear algebra problem. Starting with the polynomials $g_1,g_2,\ldots,g_n,g_{n+1}=f$ one may ask if it is possible to find an …

Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us as …

Let $k$ be an arbitrary closed field (of arbitrary characteristic). Assume that we have a short exact sequence of k-algebraic abelian connected groups $$ 1\rightarrow K\rightarrow G \rightarrow H\rig …