I am reading Beauville's chapter IX on Elliptic surfaces. Let $S$ be a minimal elliptic surface with $\kappa=1$ and $p:S\rightarrow C$ be the elliptic fibration. We know $K^2=0$ …

Let $X$ be a smooth projective surface not contained in $\mathbb{P}^3$. Is there any known condition on $X$ under which I can embed it into $\mathbb{P}^3$ such that the its image …

I was wondering about a generalization of the following property of surfaces of general type. Let $X$ be a smooth projective surface of general type. Then there is no pencil of ra …

Let $C$ be a curve in $\mathbb{P}^3$, possibly non-reduced. Assume, there exists a smooth surface in $\mathbb{P}^3$ containing $C$. Is it true that for $d \gg 0$, a generic element …

Let $C$ be a projective curve (over an algebraically closed field) of genus $\geq 1$. Let $S = C \times C$. By normalisation we have a ramified cover $C \to \mathbb{P}^1$ and so a …

The following got no answer on mathstackexchange. I believe it not to be hard, but maybe it is a little specialized? All varieties will be over $\mathbb{C}$ and projective unless …

I would like to know if there can be some kind of classification of normal rational surfaces with Gorenstein singularities, such that their canonical divisor is effective. Additi …

It is easy to show using birational factorization that the only curves on a surface which can be contracted to get an algebraic, smooth surface are smooth $(-1)$-curves. Furthermor …

Main question. Does there exist a smooth projective morphism $X\to$ Spec $\mathbf Z$ of relative dimension two such that the canonical sheaf $\omega_{X_{\mathbf Q}}$ of the generic …

Let $X$ be a projective surface defined over a field $k$ of characteristic $0$, and let $G$ be a finite group acting biregularly on $X$. Assuming that $X$ is rational over $k$, is …

How many lines in $\mathbf{P}^5$ passing through a fixed point $p$ meet in at least two points a fixed smooth surface $S$ given by the intersection of three quadrics? Or equivalen …

In his Algebraic surfaces book, Beauville gives a result allowing one to "absorb ramification" for certain maps (see below). There are also something similar one can do with number …

Let $A=\mathbb{C}^2/\Lambda^2$, where $\Lambda=\mathbb{Z}+i\mathbb{Z}$, be an abelian surface. Then every body knows that the resolution of the quotient $A/<\pm>$ is a K3 surfac …

Let $X$ be a projective normal surface over $\mathbb{C}$. In this related question it is stated as soon as $X$ is smooth any vector bundle defined on the compliment of a codimensi …

Recently I posted an "announcement" on arxiv where I said something to the effect of "this is the first example we know where contracting (a tree of) rational curves from a non-sin …