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Let $D$ be a divisor on a (complex) K3 surface. Suppose $D^2\geq0$. In general, $D$ is nef if $D\cdot C\geq0$ for all irreducible curves on the surface. Is it sufficient in our case to check this for …

Let $S$ be a complex algebraic (smooth) surface and $\widetilde{S}$ be the blowup of $S$ at a point $p\in S$. I would like to understand the statement: As a topological manifold, $\widetilde{S}$ …

It is a basic question and I would be happy to be directed to some reference for it. Let $f\colon X\to Y$ be a finite branched cover of smooth projective varieties, $M$ a line bundle on $Y$ and $L=f^ …

Let $k$ be a field of characteristic zero and $X,Y$ be integral schemes of finite type. Assume we have a dominant morphism $\pi\colon X\to Y$. Then we know that $\pi$ is generically smooth (i.e. on a …

I apologise for the basic question; I am reading Huybrecht's Lecture Notes on K3 surfaces, and on p.257 it is mentioned an example of K3 surface with infinitely many smooth rational curves. Precisely, …

Let $C$ be a general curve, say of even genus $g=2s$. Then $C$ has finitely many pencils $|L|$ of degree $\deg L=[g+3]/2=s+1$. Choose one such. The residual series is of degree $\deg(K_C-L)=3s-3$. I …

Let $S$ be an Enriques surface and $C\subset S$ a smooth irreducible curve of genus $g$. Consider the condition $$\omega_C\simeq N_{C/S}$$ For example, when $g=1$ then $\omega_C=\mathcal{O}_C$ and th …

Everything over $\Bbb{C}$. Say we have a smooth curve $C$ of genus $10$ which is a double cover of a smooth plane cubic curve. Therefore $C$ admits a 1-dimensional family of pencils of degree 4 (arisi …

By the Brill-Noether Theorem, a general curve $C$ of genus $g\geq2$ has maximal Clifford index $\lfloor \frac{g-1}{2}\rfloor$. Hence a very naive question is: (Q1) Is a curve with maximal Clifford in …

I have often seen the assertion that for a smooth plane curve $C$ of degree $d$ the gonality of $C$ is $d-1$ and each gonality pencil is obtained by projection from a point of $C$ onto a line. (let m …

Let $X$ be a smooth projective complex K3 surface and $L, D$ two effective divisors, $L^2\geq0$ and $D^2\geq0$. (Q1). do we have $L\cdot D\geq0$ ? If either one has positive self-intersection, the …

Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$. We shall say: $D$ is connected if its support is connected. $D$ is numerically connected if for any non-trivial effective decompo …

Let $X$ be a K3 surface and $D$ an effective divisor such that $h^0(D)\geq2$ and $h^1(D)=0$. Is this enough to show that $D$ is connected? Any reference would also be appreciated (I looked in Sain …

Under which circumstances is the Clifford index of a curve computed by pencils?

Let $X$ be a (smooth complex algebraic) surface. Suppose $\theta$ is an automorphism of order $2$ of $X$, such that its fixed locus is a disjoint union of smooth curves. I am trying to prove that the …