# Tagged Questions

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### geometric motivation for spaces with functions

Let $k$ be a field. A space with functions over $k$ is topological space X together with a family $O_X$ of k-subalgebras $O_X(U)\subseteq Map(U,k)$ for every open set $U$ that satisfy a) If ...
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### What is $T^*(\mathbb{A}^1)$?

Let $\mathbb{A}^1$ be the affine line and $T^*(\mathbb{A}^1)$ the contangent space of $\mathbb{A}^1$. What is $T^*(\mathbb{A}^1)$? Is $T^*(\mathbb{A}^1) = \mathbb{A}^2$? Thank you very much.
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### Application of Riemann Roch

I have read that thanks to Riemann Roch theorem, if get $\Sigma$ a compact Riemann Surface of genus $g$ there exists a conformal branch covering $\phi: \Sigma \rightarrow S^2$ of degree less than ...
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### Book Recommend Differential Geometry of Algebraic Manifolds

I just want to study Differential Geometry of Algebraic Manifolds. but I can`t find a book about that. Is there any good book for studying Differential Geometry of Algebraic Manifolds??
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### Fano manifolds and positive Ricci curvature.

I already know that if a Kahler manifold, $M$, is Fano, then $M$ has Kahler metrics with positive Ricci curvature. What about the converse? If $Ric(M)$ is positive, does this mean that $-K_M$ is ...
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### Intersection of two open sets in the projective plane

I want to compute the cohomology groups of the real projective plane, $P^2$, using Mayer Vietoris exact sequence. Now $H^0(P^2)=\mathbb{R}$, $H^2(P^2)=0$ being $P$ not orientable, so my problem really ...
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### Degree of polynomial seen as a smooth map

I need some help with a part of an exercise. Let $P$ be a real polynomial of degree $d$, seen as a map $P:\mathbb{R}\rightarrow\mathbb{R}$. Prove that if $d$ is even then the degree of $P$, $degP$, ...