Harry
Reputation
1,326
Next privilege 2,000 Rep.
 Jan 18 revised Is the relative rank function with respect to an ample line bundle non-decreasing added 516 characters in body Jan 18 asked Is the relative rank function with respect to an ample line bundle non-decreasing Jan 13 comment Are there moduli spaces of higher-dimensional varieties When I say moduli stack I mean a Deligne-Mumford stack solving the moduli problem. Sorry for not saying that explicitly. Jan 13 asked Are there moduli spaces of higher-dimensional varieties Jan 8 accepted Pulling-back a divisor and reducing it Jan 8 asked Pulling-back a divisor and reducing it Jan 5 accepted Can a ring of integers contain a $2$-dimensional noetherian normal integral domain? Jan 4 asked Can a ring of integers contain a $2$-dimensional noetherian normal integral domain? Dec 23 accepted Is the complement of an ample divisor always affine Dec 23 asked Is the complement of an ample divisor always affine Dec 23 comment Relationship between complex number and vectors If $z$ is a complex number, then $(Re(z),Im(z))$ is a vector in $\mathbf{R}^2$. Dec 22 comment Which algebraic variety can become a algebraic group? The Euler characteristic of an algebraic group should be zero. This follows from the trace formula. In fact, for any non-trivial element $a$, the translation $t_a$ by $a$ has no fixed points. By the trace formula, the trace of $t_a$ on the cohomology of your algebraic group should be zero. The latter equals the euler characteristic. In particular, this shows that in dimension $1$, the genus has to be zero because the Euler characteristic equals $2g-2$. Moreover, note that the Euler characteristic of a torus $\mathbf C^g/Lambda$ is indeed zero thus it all works out. Dec 22 comment Intuition behind isomorphism of algebraic varieties Say $V$ and $W$ are isomorphic algebraic sets. Intuitively, it should mean that the set of equations defining $V$ can be transformed into the set of equations defining $W$. For instance, consider two curves $f(x,y) =0$ and $g(x,y)=0$ in $\mathbf{A}^2$. These being isomorphic means that there is an isomorphism $k[x,y]/(f)\to k[x,y]/(g)$. Dec 22 accepted Does this equation have integer solutions Dec 21 asked Does this equation have integer solutions Dec 21 accepted Why is the rank of $f_\ast L$ the degree of $f$ Dec 21 comment If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$ Thank you! What an amazing result by Fujita! Dec 21 asked Why is the rank of $f_\ast L$ the degree of $f$ Dec 21 comment If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$ Thank you very much! Your example did convince me that this is not going to be true. A counterexample shouldn't be hard to find, but it will be cumbersome. Dec 21 revised If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$ added 60 characters in body