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bio website math.washington.edu/~kovacs
location Seattle, WA
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visits member for 4 years, 8 months
seen 22 hours ago
I am an algebraic geometer.

22h
comment how does one understand GRR? (Grothendieck Riemann Roch)
@GregorBruns: You are absolutely correct, I left out a term (I had curves in my mind), but I now corrected it. Thanks for spotting the omission!
22h
revised how does one understand GRR? (Grothendieck Riemann Roch)
corrected the exponential Chern character of L and the consequences of this correction.
1d
comment On the number of irreducible components of an exceptional divisor
@JeskoHüttenhain:It's actually not surprising. Think about the case of a quadric cone and a ruling. If you blow up the line with the reduced scheme structure, you get the same as if you blew up the point, because it is not a Cartier divisor. But if you blow up the double line, then nothing happens, because you blew up a Cartier divisor. Similar things can happen if X is smooth, but Z is singular.
2d
answered On the number of irreducible components of an exceptional divisor
May
6
comment Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?
On a curve a non-empty subset is open if and only if its complement is finite. $X_0$ contains the open subset $U_0$ and hence the complement of $X_0$ is contained in the complement of $U_0$, which is finite. OK?
May
4
answered Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?
Apr
28
comment Determine existence of irreducible variety in given homology class
Also, you would probably want to ask this differently. Say, not for a general homology class, but say a class in $\mathrm{Pic}\, X$. Otherwise, besides solving the Schottky problem you are also in danger of proving the Hodge Conjecture.
Apr
28
comment Simple example of a ring which is normal but not CM
@VA: Nice reference!
Apr
14
comment Cohen-Macaulayness of the direct image of the canonical sheaf
Just to extend on Karl's excellent answer: a slightly more general situation (that is, not just for $\omega_X$) is handled in Kollár's recent book (goo.gl/JE2CyM). Look at section 2.5.In particular, Theorem 2.74 gives exactly what Karl is saying with the substitution $\mathscr G=\omega_X$.
Apr
12
answered Question about divisors and its images
Mar
30
answered Are there any Algebraic Geometry Theorems that were proved using Combinatorics?
Mar
16
answered Vanishing of the top Chern class of a vector bundle
Mar
3
revised Global section of very ample line bundles and its value on stalks
added 47 characters in body
Mar
3
comment Global section of very ample line bundles and its value on stalks
Karl, you are right. I was only thinking about 0 or not zero.
Mar
3
comment Global section of very ample line bundles and its value on stalks
To get a surjective map to a direct sum you need to get a single element that maps to the various choices. If the same point is repeated, then on the right hand side you can choose an element which has a component that is zero at that point and another component that is non-zero. Then to get surjectivity you would have to find a single section on the left hand side that maps to this element, but it could only map onto one of those components.
Mar
3
answered Global section of very ample line bundles and its value on stalks
Mar
2
comment Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial
(con't-ed) This may seem a trivial point to you now, but it can lead to disaster if you don't keep your notions in order. As an exercise compare the push forwards $f_*K_f$ and $f_*\omega_{X/B}$...
Mar
2
comment Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial
@Vesselin: of course you did. I didn't think you meant the tensor power, but no one can know what you would have written if you meant something else than what you have actually written! And just to be a bit nitpicking, there is no such thing as the self-intersection of a line bundle. That's my point. Cycles have self-intersections and line bundles have associated divisors which are cycles. (OK, you could define self-intersection of a line bundle as the self-intersection of the associated divisor, but that's besides the point)...
Mar
2
answered Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial
Mar
2
comment Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial
Please don't write $K^2=\omega^2$!! One of them is a divisor, the other is a sheaf. There is a reason these notions are distinguished. $\omega^2$ is the line bundle corresponding to the divisor $2K$ and not the number $K^2$. Of course, what you have in mind is different, but then why not write down what you have in mind? It's not that hard. You could have written for instance that $K^2=c_1(\omega)^2$...