Search Results

The exercises in the early chapters of the book by Arbarello Cornalba Griffiths and Harris are very interesting. The book itself is a second course but the early chapters and execises are a recap with …

Since my comment was too cryptic, I will spell it out as an answer. A polarization on an abelian variety $A$ is an ample divisor $D$ (modulo linear equivalence). A polarization is principal if the s …

Extending Rita's example, if $X$ is, say, a double cover of a curve of genus $3$, then $X$ can have arbitrarily large genus and it has gonality (at most) $6$. Moreover it has infinitely many $g^1_6$ ( …

Joking aside... Let $x: X \to \mathbb{P}^1$ be of degree $2$ and $\sigma: X \to X$ be an automorphism. Consider $f: X \to \mathbb{P}^1\times\mathbb{P}^1, f=(x,x\circ\sigma)$. If $f$ is injective, then …

As mentioned by others, the first proof uses a 3-descent, namely the map $E(K)/3E(K) \to H^1(K,E[3])$ (where $E$ is the elliptic curve and $K = \mathbb{C}(t)$) obtained by taking (Galois) cohomology o …

By (a possible) definition, a principal polarization on an abelian surface is a curve with self-intersection 2. So, if smooth, it is a genus two curve and the abelian surface is a jacobian. You have t …

Are the $D_i$ supposed to be points? This is not made clear in the question. If the $D_i$ are points and you exclude Jack's example where they are all the same, then the answer is no for the general c …