JGuy

Hello. Does anyone know where I can know about the hodge diamond of projective curves?

be right back I will be back later.

Excuse me. I want to ask, so $H$ is given by $V(h) \cap X$. Look at the homogeneous polynomial h, if we pick an affine chart and set one of projective coordinate equal to 1. If we convert back to Projective coordinates we get rational function $s$ on $X$ such that $div(s)_X = H + \text{other things}$?

yes

If X is a variety and $D$ is a divisor then for $m$ big enough we have $mH - D$ is very amply, so there exists effective divisor $D^{\prime}$ such that $mH - D \sim D^{\prime}$?

@TedShifrin Yeah, so we we get the above. Ty!

What does hyperplane section section of a variety X means? Does it mean it is given by a single equation? If $H$ is hyperplane section in $X$, does this mean that $H = V(h) \cap X$?

Hello

The reason I want this is because I want to construct some metric on Y

given a smooth family of maps $X \rightarrow (0,\infty)$ I want to construct a smooth family of maps $Y \rightarrow (0,\infty)$

I want to construct a smooth family on $Y$

@leslietownes 1 moment I will explain

so $p_{\alpha}$ is a smooth family on X

1 sec reading

@leslietownes wait what?

@leslietownes do you agree?

then we can define $p_{\alpha}^{\prime} : X \rightarrow (0,\infty)$ I think this is also a family of smooth maps, do you agree?

surjection will allow that given any element $y \in Y$ we have at least one $x_y$ that maps to $y$ we can use axiom of choice to make a consistent choice of $x_y$ that maps to $y$.

@leslietownes If I have a collection of $C^{\infty}$ families on $Y$ denoted as $p_{\alpha} : Y \rightarrow (0,\infty)$ then I want a collection of $C^{\infty}$ families on $X$.

@leslietownes I wrote it in the wrong direction

Suppose we have a surjective map $\phi : X \rightarrow Y$. Let us say we have collection of $C^{\infty}$ maps on $X$ denoted as ${ p_{\alpha} : X \rightarrow (0,\infty) }$. We can use axiom of choice to define a collection of $C^{\infty}$ maps on Y right? That is a collection ${ p^{\prime}_{\alpha} : Y \rightarrow (0,\infty) }$.

Hello