# Tagged Questions

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### On Severi's definition of the complementary correspondence

In Weil's short note entitled "On the Riemann hypothesis in function-fields" he mentions the notion of the complementary correspondence associated to a given correspondence $T:C\rightarrow C$ where ...
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### Effective cone of C x C where C is a Fermat curve

A search of the literature reveals that for a curve $C$ of genus $\geq 2$, determining the effective cone of $C \times C$ is hard. My question is this: do we know a single example of a curve $C$ of ...
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### Uniqueness of lifting of very ample line bundle on smooth proper surfaces over DVR

Let $R$ be a complete Henselian discrete valuation ring of characteristic 0, $X_R$ be a surface smooth, proper and flat over $R$. Assume that the residue field $k$ of $R$ is algebraically closed of ...
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### The central fiber of this family of surfaces?

I have a question on a description of a central fiber of the following family of surfaces. Let $B,C \subset \mathbb{P}^2$ be smooth curve of degree $2d$ and $d$ respectively. Let's consider the ...
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### Existence of smooth surfaces containing a curve

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 of $I_d(C)$ defines ...
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### Producing $(-2)$ curves on a smooth surface

We know that blowing up a point on a surface produces a $(-1)$ curve. Is there any such standard techniques to produce $(-2)$ curves in a smooth surface?
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### How to tell if a second-order curve goes below the $x$ axis?

Suppose we have a second-order curve in general form: (1) $a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0$. I'd like to know if there is a simple condition that ensures that the curve ...
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### Equation for simple Jacobian of a genus two curve

Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$. Can we write down an explicit equation for the abelian surface $J$? I know $X$ can be ...
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### is there some condition I can impose on families of curves on a surface such that the second Ext between the ideal sheaf and the structure sheaf is zero?

(the title got out of hand) Say I have a surface $X$, then I also have M, the Hilbert scheme of curves and points on X. This can be seen as a moduli space of quotients $O_X \to O_Z$. If $I_Z$ is the ...
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### difference of curve classes

Let $X$ be a smooth protective variety, or just a smooth Kahler manifold. Is it possible to have two curves $C_1$ and $C_2$ in $X$ such that their difference in $H_2(X,\mathbb{Z})$ is a non-trivial ...
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### How to find the action of an automorphism on the 27 lines on a cubic surface?

Assume $C\subset \mathbb{P}^2$ is a smooth cubic curve. Then there is a cyclic triple cover $\pi: S\rightarrow \mathbb{P}^2$ ramified in $C$. Let $\sigma$ be the covering automorphism (sheet ...
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### Divisorial contraction: when is the image an algebraic space or a stack?

Let $X$ be a smooth projective surface (in the category of varieties, or schemes), and let $C\subset X$ be a curve (a priori not irreducible, but the irreducible case in itself is already ...
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### Pencil of lines and degree $d$ curve in $\mathbb{CP}^2$

Question. Let $C$ be a generic smooth curve of degree $d$ in $\mathbb{CP}^2$, and let $P$ be an arbitrary point away from this curve. How many lines are there through point $P$ that are tangent, or ...
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### Extending birational isomorphisms between planar curves to the P^2

Let $k$ be a field and let $C,D$ be two integral curves in $\mathbb{P}^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $\mathbb{P}^2_k$. To be precise, does there exist a ...
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### Defining ideals for rational curves in space

A rational normal curve $C_d\subset\mathbb{P}^d$ is defined by quadrics. I guess, for the generic projection $\mathbb{P}^d\stackrel{\pi}{\to}\mathbb{P}^n$ the image $\pi(C_d)$ is still defined by ...
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### 'Reference' request: Program to work with cyclic quotient singularities.

I'm looking for program code to deal with cyclic quotient singularities on normal surfaces. In particular, at the moment I need that given a singularity $p$ like $p=\frac{1}{n}(1,a)$ the code ...
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### Example of cone of numerically effective curves which is not polyhedral

I think I have seen more than one reference in which the cone of numerically effective curves can be 'not polyhedral', i.e. with an infinite number of extremal rays I cannot remember where I read ...
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### Numerically negative exceptional divisor on a surface.

Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection ...
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What is it known now about Nagata's conjecture and Seshadri constant (http://en.wikipedia.org/wiki/Nagata%27s_conjecture_on_curves and http://en.wikipedia.org/wiki/Seshadri_constant) for toric ...
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### Does each finite morphism of curves have a model whose minimal resolution is semi-stable

Let $\pi:Y\to X$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$. Question. Does there exist a finite field extension $L/K$ and a regular model ...
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### Log resolutions on surfaces and 3-folds in characteristic p

If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the ...
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### Del pezzo surfaces in positive characteristic

For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second ...
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### The exceptional locus of a minimal resolution of singularities

Let X be a surface. (A surface is an excellent integral normal separated 2-dimensional scheme.) Let $\psi:Y\longrightarrow X$ be a minimal resolution of singularities and let $E$ be an irreducible ...
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### Why is it a non-basepoint?

In the proof of the Castelnuovo theorem for curves in $\mathbb{P}^3$ (Hartshorne IV, 6.4.) the following is done: One considers a smooth, complete curve $C$ in the projective space $\mathbb{P}^3$ over ...
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### Global sections of a linear system

Recently, following Beauville's book (exercises iv.(1),(2)) I have been working on Hirzebruch surfaces (from the algebraic geometry point of view) and I had to compute the space of global sections of ...
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### The number of singular fibres of a semi-stable arithmetic surface over \Z

This is an arithmetic follow-up to my previous question Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres Let $k$ be an algebraically closed field and let ...
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### Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres

This question is based on Beauville's article in Szpiro's asterisque Seminaire sur les pinceaux de courbes de genre au moins deux from 1986. We will work over the complex numbers $\mathbf{C}$. Let ...
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### Surfaces containing curves of arbitrarily negative self-intersection

Olivier Wittenberg and I are curious about the following : Let $S$ be a smooth projective complex surface. Are the self-intersection numbers of integral curves on $S$ always bounded below ? Or can ...
Not much is known about vector bundles on $\mathbb{P}^2$ but I wonder if the following is a tractable question: If $E,E'$ are non-isomorphic vector bundles on $\mathbb{P}^2$, then is there always a ...