surfaces
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Problem 32：Resolution of DuVal Singularities Part 1.
Exercise: Resolve the 5 forms of DuVal Singularities (Part 1/5: $A_n$ singularities). The Minimal Model Program…
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Problem 31: Hartshorne V.1.3.
Hartshorne Exercise V.1.3. Recall that the arithmetic genus of a projective scheme D of dimension 1…
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Problem 21: Beauville II.20
Beauville Exercise II.20. Let C be curve on a smooth, projective surface S. Show that there…
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Problem 15: Hartshorne V.1.4
Hartshorne Exercise V.1.4. (a) If a surface $X$ of degree $d$ in $\PP{k}{3}$ contains a straight…
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Problem 14: Hartshorne V.1.11
Hartshorne Exercise V.1.11.In this problem, we assume that $X$ is a surface for which $\op{Num}X$ is…
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Problem 13: Hartshorne V.1.9
Hartshorne Exercise V.1.9. (a) If $H$ is an ample divisor on a surface $X$, and $D$…
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Problem 12: Hartshorne V.3.3
Hartshorne Exercise V.3.3. Let $\pi:\tilde X\ra X$ be a monoidal transformation, and let $D$ be a…
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Problem 11: Hartshorne V.3.2
Hartshorne Exercise V.3.2. Let $C$ and $D$ be curves on a surface $X$, meeting at a…
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Problem 10: Hartshorne V.1.6
Hartshorne Exercise V.1.5 (a) If $C$ is a curve of genus $g$, show that the diagonal…
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Problem 9: Hartshorne V.1.5
Hartshorne Exercise V.1.5 (a) If $X$ is a surface of degree $d$ in $\PP{k}{3}$, then $K^2=d(d-4)^2$.…
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Problem 8: Hartshorne V.1.2
Hartshorne Exercise V.1.2 Let $H$ be a very ample divisor on a surface $X$, corresponding to…
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Problem 2: Hartshorne Exercise IV.3.3 and some generalizations of Problem 1
We first make some interesting generalizations to the previous post, problem 1. Recall that we showed…
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Problem 1: Show that the blowup of $\Bbb P_k^2$ at up to 8 points can be Fano.
Define inductively a sequence of blowups $\varepsilon_n:S_n\ra S_{n-1}$, where $\varepsilon_n:S_n\ra S_{n-1}$ is the blowup of $S_{n-1}$…