# Tagged Questions

**6**

votes

**3**answers

215 views

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

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 ...

**4**

votes

**2**answers

231 views

### Varieties with big anti-canonical divisor

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 ...

**5**

votes

**1**answer

385 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 $p_1,...,p_n$. ...

**1**

vote

**1**answer

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 ...

**3**

votes

**3**answers

178 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. ...

**3**

votes

**0**answers

182 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 ...

**3**

votes

**2**answers

212 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 ...

**3**

votes

**1**answer

302 views

**7**

votes

**2**answers

386 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 ...

**4**

votes

**1**answer

288 views

### Property of singularity

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 ...

**4**

votes

**2**answers

549 views

### Vanishing associated to a resolution of singularities

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 ...