Timeline for Linear systems on moduli spaces of stable maps
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Jun 7, 2017 at 21:17 | vote | accept | CommunityBot | ||
| Jun 7, 2017 at 19:27 | answer | added | Jason Starr | timeline score: 1 | |
| Jun 7, 2017 at 18:19 | comment | added | user97096 | I agree with yuor last comment. Anyway I am confused about the fact that any contraction is induced by a complete linear system. In my second comment the isomorphism $f:\mathbb{P}^2\rightarrow Y$ (which in particular is a contraction) is induced by a proper sublinear system of projective dimension $4$ of $|\mathcal{O}_{\mathbb{P}^2}(2)|$: the complete linear system of plane conics which has projective dimension $5$. | |
| Jun 7, 2017 at 18:16 | comment | added | user97096 | Consider $f_{I}:\overline{M}_{0,n}(\mathbb{P}^r,d)\rightarrow \overline{M}_{0,n}\subset\mathbb{P}^N$. Then $D_{f_I}$ is linearly equivalent to the pull-back of a hyperplane section of $\mathbb{P}^N$ restricted to $\overline{M}_{0,n}$, My question could be reformulated as follows: Do we have that $N = dim(H^0(\overline{M}_{0,n}(\mathbb{P}^r,d),D_{f_I}))-1$ ? | |
| Jun 7, 2017 at 16:21 | comment | added | Jason Starr | I do not understand your question. What is the divisor "associated to $f_I$"? If you mean the pullback of an ample divisor by $f_I$, then my previous comment answers your question. | |
| Jun 7, 2017 at 16:14 | comment | added | user97096 | Now I see where the confusion comes from. I edited the question. Sorry for the misunderstanding. | |
| Jun 7, 2017 at 16:12 | history | edited | user97096 | CC BY-SA 3.0 |
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| Jun 7, 2017 at 16:00 | comment | added | Jason Starr | If $f:X\to Y$ is a contraction, if $(\mathcal{L},V\to H^0(Y,\mathcal{L}))$ is a complete linear system on $Y$, then $(f^*\mathcal{L},V\to H^0(X,\mathcal{L}))$ is a complete linear system on $X$. This follows from the projection formula, $f_*f^*\mathcal{L} \cong \mathcal{L}\otimes_{\mathcal{O}_Y} f_*\mathcal{O}_X$ and the fact that $\mathcal{O}_Y\to f_*\mathcal{O}_X$ is an isomorphism. | |
| Jun 7, 2017 at 15:53 | comment | added | user97096 | From a general point $p\in\mathbb{P}^5$ not $p\in\mathbb{P}^4$. | |
| Jun 7, 2017 at 15:47 | comment | added | user97096 | Ok. But consider the morphism $v:\mathbb{P}^2\rightarrow V\subset\mathbb{P}^5$ induced by the complete linear system of plane conics. Here $V$ is the Veronese surface. The surface $V$ can be projected isomorphically onto a surface $Y\subset\mathbb{P}^4$ from a general point $p\in\mathbb{P}^4$. The composition of $v$ and the projection gives an isomorphism $f:\mathbb{P}^2\rightarrow Y$ which satisfies your definition of contraction but by construction the linear system inducing $f$ is not complete. Perhaps I am missing something. | |
| Jun 7, 2017 at 15:37 | comment | added | Jason Starr | For normal schemes, a contraction is a proper, finitely presented morphism $f:X\to Y$ such that the induced morphism $f^\#:\mathcal{O}_Y\to f_*\mathcal{O}_X$ is an isomorphism. Using Stein factorization, it suffices to prove that every geometric generic fiber is irreducible and reduced. | |
| Jun 7, 2017 at 15:31 | comment | added | user97096 | Thanks a lot. Just one last question. What do you mean exactly by contratction here? If I got it right you want the fibers to be connected. Do we need to assume that the fibers are of positive dimension? What if we have a birational morphism $f:X\rightarrow Y$ of normal projective varieties? I think that if $f$ is birational we must assume that it is not an isomorphism. | |
| Jun 7, 2017 at 11:23 | comment | added | Jason Starr | One final comment: we do know how to completely describe the nef cones of the spaces $\overline{M}_{0,n}(\mathbb{P}^r,d)$ in terms of the nef cones of the moduli space $\overline{M}_{0,m}$, cf. my papers with Izzet Coskun and Joe Harris. We also know the pullback maps under $f_I$ and $g$ relative to the tautological generators of the Picard groups. So it should be straightforward to find an explicit divisor class on $\overline{M}_{0,n}(\mathbb{P}^r,d)$ that is the pullback under $f_I$, resp., $g$, of an ample divisor class. | |
| Jun 7, 2017 at 10:49 | comment | added | Jason Starr | A similar argument applies for the forgetful morphism $g$. Since the source and target are normal, to prove that $g$ is a contraction it suffices to prove that the geometric generic fiber is connected. The general fiber is birational to the projective space $\mathbb{P} H^0(\mathbb{P}^1,\mathcal{O}(d))^{\oplus (r+1)}$. | |
| Jun 7, 2017 at 10:31 | comment | added | Jason Starr | Every contraction between normal, projective varieties comes from a complete linear system. The spaces $\overline{M}_{0,n}(\mathbb{P}^r,d)$ are projective coarse moduli spaces of the algebraic stacks with finite diagonal $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^r,d)$. These algebraic stacks are smooth (cf. Kontsevich's "Enumeration of Rational Curves via Torus Actions" or other sources). Thus the coarse moduli spaces are normal. The forgetful morphisms are contractions: the geometric fibers are integral schemes parameterizing $|I|$-tuples of points on a connected, projective curve. | |
| Jun 7, 2017 at 9:24 | history | asked | user97096 | CC BY-SA 3.0 |