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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 …

Q1: What is the simplest example of two non-isomorphic fields $L$ and $K$ of characteristic $0$ such that $L(x)\simeq K(x)$ (here $x$ is an indeterminate)? Q2: Do we have a sufficient criterion for …

So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_0$ at a prime $p$ is given by $Z_p(T)=1-T$ (a polynomia …

Let $X_0$ be a smooth projective variety over $\mathbb{C}$ and let $\Theta_{X_0}$ be the locally free sheaf of $O_{X_0}$-module corresponding to tangent space of $X_0$. Goal: To find a sufficient co …

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 $X\subseteq\mathbb{P}^n(\mathbf{C})$ 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 …

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 $V$ be a smooth projective variety defined over $\mathbf{Q}$ and denote by $$ \omega: H_{dR}^*(V,\mathbf{Q})){\otimes_{\mathbf{Q}}}\mathbf{C}\rightarrow H_{B}^*(V,\mathbf{Q})\otimes_{\mathbf{Q}}\m …

Let $X$ be a smooth projective curve over a field $k$. We let $\omega$ be the canonical line bundle of $X$ and we denote by $F$ the field of $k$-valued rational functions on $X$. (1) When $k$ is alg …

Let $\pi:\mathcal{E}\rightarrow U$ be a family of elliptic curves with level $n$ structure (in the sense of Deligne-Rapoport) where $U\subseteq C$ is some (non-empty) Zariski open set of a smooth comp …

So this question is directly related to a comment made by David Mumford in his Lecture 1 given at U. Michigan in 1974 entitled: What is a curve and how explicitly can we describe them ? Mumford cla …

Let $H$ be a smooth projective hypersurface in $\mathbb{P}^n(\mathbb{C})$ where $n\geq 3$. Then by the Lefschetz hyperplane theorem we have that $H^1(H,\mathbb{C})= H^1(\mathbb{P}^n(\mathbb{C}),\mathb …

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 …

Let $X$ and $Y$ be smooth quasi-projective varieties defined over $\mathbf{C}$ and let $$ f:X(\mathbf{C})\rightarrow Y(\mathbf{C}) $$ be a holomorphic map (not necessarily regular=algebraic). Then it …

So let $X$ be a projective hypersurface inside $\mathbb{P}_{\mathbb{Z}}^n$ of degree $d$. Assume that (a) $X(\mathbb{C})$ and $X(\overline{\mathbb{F}}_p)$ are irreducible, (b) and that $X(\mathbb{C …