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bio website math.sunysb.edu/~jstarr/…
location Stony Brook University, Stony Brook, NY USA
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visits member for 5 years, 6 months
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6h
comment Variation of a very ample line bundle along a flat family
No, that is not true. You should look up the definition of "linearly normal" and "projectively normal".
Apr
21
comment McDuff's classification of symplectic manifolds
What about the projective completion of a degree-$1$ complex line bundle over a genus-$g$ Riemann surface? This is a symplectic manifold that is not symplectomorphic to a blowing up of $\mathbb{C}P^2$, since the fundamental group is wrong.
Apr
20
comment Surjectivity of trace map
Dear all, I only just now realized that when the OP asks about "characteristic 0", probably that means that the fraction field should be characteristic 0 (i.e., mixed characteristic), rather than that the residue field should be characteristic 0 (equicharacteristic $0$). Others have given examples in mixed characteristic, so I will leave my answer as it is.
Apr
20
comment Is $K^0(X)\to K_0(X)$ monomorphic for a noetherian scheme $X$?
@ZhaotingWei: "... do we have the injection result if we assume that the scheme $X$ is reduced?" Unfortunately not. There are examples as above where $X$ is projective, and both $X$ and $X_{\text{red}}$ are local complete intersection schemes over a field $k$. Let $i:X\to \mathbb{P}^n_k$ be a closed immersion. Let $\nu: Y \to \mathbb{P}^n_k$ be the blowing up of the image of $i$. Using the formula for K-theory of a blowing up, $K^0(Y)\to K_0(Y)$ is not a monomorphism, but now $Y$ is reduced.
Apr
20
comment Surjectivity of trace map
@Hebe: "But what if the characteristic is $0$?" If the degree $d$ of $K/F$ is invertible in $R$, then the trace of $1/d$ equals $1$. Thus the trace map is surjective.
Apr
20
comment Stabilisers of group actions
@DanielLoughran: That is literally the algebraic group of smallest dimension that admits a one-parameter family of closed subgroups (or rather, a closed subgroup scheme $\Gamma$ over the base $\Delta$) that specializes from $\mu_2$ to $\mu_1$. Now you just work out what is the quotient of $\Delta\times G$ by $\Gamma$.
Apr
20
answered Surjectivity of trace map
Apr
19
revised Stabilisers of group actions
added 176 characters in body
Apr
19
answered Stabilisers of group actions
Apr
17
comment Are universally catenary equidimensional local rings Cohen-Macaulay?
Do you want to assume that your ring is reduced and seminormal?
Apr
17
comment Dominating affine varieties over $k$ with affine smooth varieties over $k$
Just define $\widetilde{X}$ to be the basic open $D(f)$ in $X$ for $f$ any nonzero polynomial in the appropriate Fitting ideal of $\Omega_{X/K}$.
Apr
17
answered Is $K^0(X)\to K_0(X)$ monomorphic for a noetherian scheme $X$?
Apr
15
comment Simply connected Kahler manifold without any effective divisor
There is a Torelli theorem for Kaehler K3 surfaces. So, for a point in the period domain parameterizing $[H^{2,0}(X)] \in \mathbb{P}H^2(X,\mathbb{C})$ such that $H^{2,0}(X)$ is not orthogonal in $H^2(X,\mathbb{C})$ to any given (nonzero) integer point of $H^2(X,\mathbb{Z})$, there is a corresponding Kaehler K3 surface $X$ that has no nonzero, integral $(1,1)$-classes. Thus, these K3 surfaces have no effective divisors.
Apr
15
comment Examples of surface automorphisms with no periodic points
@J.C.Ottem: We both made the same comment (nearly) simultaneously.
Apr
15
comment Examples of surface automorphisms with no periodic points
I do not see how the paper of Oguiso addresses the question. Even if $g$ has no fixed point, nonetheless, $g\circ g$ must have a fixed point by the holomorphic Lefschetz fixed point theorem. So even if Oguiso calls this automorphism "free", that does not mean it has no periodic orbits.
Apr
14
answered Examples of surface automorphisms with no periodic points
Apr
14
comment Examples of surface automorphisms with no periodic points
Just to point out one more thing: if your automorphism $g$ fixes an ample invertible sheaf, then the action of $g$ on the complete linear system of any tensor power of the invertible sheaf admits a fixed point. Thus, there is a fixed curve inside your surface. Dynamics on curves are easy to understand, and essentially your curve must be arithmetic genus $1$ if it has no periodic orbits. In particular, that rules out Kobayashi hyperbolic surfaces.
Apr
14
comment Examples of surface automorphisms with no periodic points
By the "Wood's Hole", Atiyah-Bott, holomorphic Lefschetz fixed point formula, an automorphism $g$ of a (projective, complex) K3 surface has a fixed point so long as $g^*$ does not act on $H^0(S,\omega_S)$ as $-1$. However, even if $g^*$ does act as $-1$, then for $h=g\circ g$, then $h^*$ acts as $+1$, and thus $h=g\circ g$ has fixed points. So there will also be no examples on projective, complex K3 surfaces.
Apr
14
comment Examples of surface automorphisms with no periodic points
Every automorphism of a projective rationally connected variety over any field has a fixed point (you can prove this by reduction to the case of finite fields, where the automorphism is automatically finite order). So you cannot find examples on projective rational surfaces.
Apr
12
comment Compact locally conformal Kahler manifolds with non-zero Euler characteristic
Have you tried a blowing up at a point of Bryant's example? Blowing up of a (complex) 4-dimensional manifold increases the Euler characteristic by 3.