Timeline for Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$

7 events

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Jun 17, 2021 at 4:36	comment	added	Tabes Bridges		Positive twisting can make rational sections regular, so one interpretation is that global sections of $\mathcal F(d)$ have at least something to do with rational sections of $\mathcal F$ that blow up in a controlled way along a degree $d$ divisor $Y$. If $\mathcal F$ is a vector bundle, then from $0 \to \mathcal F \to \mathcal F(d) \to \mathcal F(d)\|_Y \to 0$ we have $H^0(\mathcal F) \subset H^0(\mathcal F(d))$ which makes this interpretation a bit more literal. OTOH if $\mathcal F$ is a torsion sheaf, one could perhaps leverage this idea by choosing $Y \supset\operatorname{supp}\mathcal F$.
Jun 16, 2021 at 15:05	answer	added	Donu Arapura		timeline score: 8
Jun 16, 2021 at 14:38	answer	added	user122276		timeline score: 2
Jun 16, 2021 at 14:15	comment	added	gigi		@FrancescoPolizzi thank you for the answer! But do you know what are, from a geometric description/point of view, elements in $H^0(T_X(d)) \setminus H^0(T_X)$?
Jun 16, 2021 at 14:08	comment	added	Francesco Polizzi		Well, if $d \geq 0$ in general we have $h^0(T_X(d)) \geq h^0(T_X)$, so the sections of $T_X(d)$ are not only vector fields. Since $H^0(T_X) \subseteq H^0(T_X(d))$, vector fields are contained here.
Jun 16, 2021 at 12:46	history	edited	gigi	CC BY-SA 4.0	added 14 characters in body
Jun 16, 2021 at 12:33	history	asked	gigi	CC BY-SA 4.0