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Let $f:X \to Y$ be a dominant morphism between two integral proper surfaces (therefore $2$-dimensional, proper $k$-schemes). …

Let $X,Y$ Noetherian integral schemes and assume we have an immersion $$i:X \to Y$$ An immersion is according to https://stacks.math.columbia.edu/tag/01IO a composition of a closed immersion $c: X \to …

Assume $R$ is a discrete valuation ring with uniformizing parameter $t$, i.e. $\mathfrak{m}_R=(t)$. We denote $\widehat{R}$ the completion of $R$ with respect to $(t)$. Let $Y$ be a flat locally Noeth …

Let consider the associated map $f: X \to Y$ of schemes $X=\operatorname{Spec}(B)$ and $Y=\operatorname{Spec}(A)$, which is well known to be finite. …

Let $f:X \to Y$ a proper surjective map between schemes $X,Y$ with additional assumption $f_*\mathcal{O}_X=\mathcal{O}_Y$. Let $y \in Y$ with a 'isolated point fiber', i.e., $f^{-1}(y)=\{x\}$ as set. … Liu's "Algebraic Geometry and Arithmetic Curves" the problem is solved in more general setting without "properness" assumption using techniques on formal schemes (Prop 4.4.2). …

In which sense is Gruson and Raynaud's relative dévissage an "extension/ generalization" of Grothendieck's "classical" dévissage concept except that the first one works in "relative" setting (ie with …

This question deals with a concrete exercise from Geomerty of Schemes by Eisenbud and Harris but also moreover the general philosophy attacking typical problems in algebraic geometry of following relative … nature: say we have a "nice" enough map (having here sloppy said something "fibration like" in mind) $f:X \to Y$ of schemes over base field $K$, and assume $Y$ (wlog we can assume it to be affine) has …

What can we say about the structure of such schemes? Topologically, algebraically,... …

My question referer to Bosch's "linear morphisms" (of schemes) $\psi: \mathbb{A}^1 _U \to \mathbb{A}^1 _U$ as desecribed below. … MY QUESTION: In Bosch's "Algebraic Geometry and Commutative Algebra" (see pages 428/ 429) there was suggested an attempt to "transfer" this formalism to schemes by do following (see below why this construction …

I have a question about following argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 159): Let $X,Y$ schemes which are finite and locally free …

Let $Y:=\mathbb{P}^1_k$ and $X$ a hyperelliptic curve. We working with the definition 4.7 from Liu's "Algebraic Geometry and Arithmetic Curves" (page 288) Namely there exist finite separable map $\ …

I have a question about a claim about classification of finite flat group schemes: https://en.wikipedia.org/wiki/Group_scheme#Finite_flat_group_schemes A fin flat group scheme $G$ is of type $(a,b)$ …

I have a curious question about an argument/hint given in following thread: https://math.stackexchange.com/questions/3062986/irreducible-smooth-proper-one-dimensional-mathbbz-schemes The OP asked if … a genus argument (interpreting it as dimension of first cohomology of $C$) on the resulting sequnce of the induced morphisms between structure sheaves to verify that $f$ is already an isomorphism of schemes …

Let $f:S→B$ be an elliptic fibration from an integral surface $S$ to integral curve $B$ . Here I use following definitions: A surface (resp. curve) is a $2$ -dim (resp. $1$-dim) proper k scheme ove …

Let $X$ be a scheme and it's Picard scheme $\underline{Pic}(X)$ exist. Denote by $\underline{Pic}^0(X)$ the connected component of the origin of the Picard scheme. My question is what the geometric in …