# Tagged Questions

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### Contracting a rational curve in a Calabi-Yau threeolfd

Let $X$ be a Calabi-Yau threefold and $C \subset X$ be a rational curve with $N_{C/X}\cong \mathcal{O}\oplus \mathcal{O}(-2)$. Can one contract the curve $C$? Assuming the answer is yes, what kind of ...
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I recently heard about the following problem: Let $X$ be a projective variety with klt singularities and such that $-K_X$ is big. Is $X$ a Mori Dream Space ? Now, $-K_X$ big if and only if $-K_X ... 1answer 384 views ### Bertini's Theorem Let$p_1,...,p_n\in\mathbb{P}^{N}$be general points. Consider the linear system$|L|$of hypersurfaces of degree$d$in$\mathbb{P}^{N}$with prescribed multiplicities$m_1,...,m_n$at$p_1,...,p_n$. ... 1answer 69 views ### Determining the desingularization from the complete local ring Suppose I have a curve$C$over a field$k$and that$p$is a singular point of$C$. Let$f : X \to C$be the desingularization of$C$at$p$. Then for each$s \in f^{-1}(p)$we have a map of local ... 3answers 176 views ### Contractibility of curves and embedding into projective space Let$f:X \to Y$be a proper surjective morphism of projective surfaces such that there exists a curve$C \subset X$for which$f|_{X\backslash C}$is an isomorphism and$f(C)$is a set of points. ... 0answers 180 views ### References for resolutions of ordinary singular points Let$X$be a$n$-dimensional complex projective algebraic variety, let us suppose that$X$has only isolated singularities. Edit: Let us say that an ordinary$m$-ple singular point is an isolated ... 2answers 207 views ### Which isolated surface singularity comes from a -5 curve? Define the surface$X$to be the total space of$\mathcal{O}_{\mathbb{P}^1}(-5)$. By contracting the exceptional curve in$X$, we get a surface with an isolated singularity. I am looking for the ... 1answer 301 views ###$A_{\infty}$singularity What kind of singularity is commonly meant by$A_{\infty}$? 2answers 380 views ### Resolution of singularities for flat families. Is there a resolution of singularities for flat families? More precisely, if$X \rightarrow \mathbb{A} ^n$is a flat map, does there exist a map$Y \rightarrow X$such that, for every$p \in ...
Let $X$ be an algebraic variety, $S \subset X$ its singular locus, and $x \in S$. Say that $x$ is good if for any neighborhood $U$ of $x$, any top differential form $\omega$ on $U \setminus S$ and ...
Let $\pi: V\to W$ be a resolution of singularities, let $E \subset V$ be the exceptional divisor, and let $F$ be a coherent sheaf such that $R^i\pi_*F=0$ for $i>0$. Can we conclude that ...