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

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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 
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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 
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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 
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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 Ktheory of a blowing up, $K^0(Y)\to K_0(Y)$ is not a monomorphism, but now $Y$ is reduced. 
Apr 20 
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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 
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Stabilisers of group actions
@DanielLoughran: That is literally the algebraic group of smallest dimension that admits a oneparameter 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 
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Are universally catenary equidimensional local rings CohenMacaulay?
Do you want to assume that your ring is reduced and seminormal? 
Apr 17 
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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 
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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 
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Examples of surface automorphisms with no periodic points
@J.C.Ottem: We both made the same comment (nearly) simultaneously. 
Apr 15 
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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 
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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 
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Examples of surface automorphisms with no periodic points
By the "Wood's Hole", AtiyahBott, 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 
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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 
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Compact locally conformal Kahler manifolds with nonzero Euler characteristic
Have you tried a blowing up at a point of Bryant's example? Blowing up of a (complex) 4dimensional manifold increases the Euler characteristic by 3. 