Ttdfjt

I just could draw the one with four arrows up parallel with one longer down also with appropriate spacing:

.

Not trivial…

The timesmathbbP^1 part is set in a zero width box.

documentclassarticle
usepackageamsmath,mathtools,amssymb,tikz-cd

begindocument

begintikzcd[row sep=3em,column sep=3em]
& M_0,4mathrlaptimesmathbbP^1
 arrow[d,"pi"',";bigguparrowbigguparrowbigguparrowbigguparrowsigma_i"]
\
B arrow[r,"kappa"'] & M_0,4
endtikzcd

enddocument

There are possibly different ways.

With this answer it is very easy. You can use the nodes for anything you want.

documentclassarticle
usepackagetikz-cd
usepackageamsmath,amssymb
begindocument
begintikzcd[column sep=2.5em,row sep=2.5em,execute at end picture=
foreach X in 1,2,3,4
draw[latex-,shorten >=1pt,shorten <=1pt] ([xshift=X*1ex-1ex]M1.south east) coordinate
(aux-X) -- 
(aux-X
]
 & |[alias=M1,text width=width("$M_0,4$")]|M_0,4times mathbbP^1
 arrow[d,"pi" swap] \
 B arrow[r,"kappa" swap] & |[alias=M2]| M_0,4 \
endtikzcd

begintikzcd[column sep=2.5em,row sep=2.5em,execute at end picture=
foreach X in 1,2,3,4
draw[latex-,shorten >=1pt,shorten <=1pt] ([xshift=X*1ex-1ex]M1.south east) coordinate
(aux-X) to[out=-90,in=80-X*10] (M2)
ifnumX=4
node[midway,right] $sigma_i$
fi;
]
 & |[alias=M1,text width=width("$M_0,4$")]|M_0,4times mathbbP^1arrow[d,"pi" swap] \
 B arrow[r,"kappa" swap] & |[alias=M2]| M_0,4 \
endtikzcd

begintikzcd[column sep=4.5em,row sep=2.5em,execute at end picture=
foreach Y in 1,2 foreach X in 1,2,3,4
-M2Y.north)
ifnumX=4
node[midway,right] $sigma_iifnumY=1 'fi$
fi;
]
 |[alias=M11,text width=width("$B'$")]|B'times mathbbP^1 
 arrow[r,shorten <=2.1em] arrow[d,"pi'" swap]
 & |[alias=M12,text width=width("$B$")]|Btimes mathbbP^1arrow[d,"pi" swap] \
 |[alias=M21]| B' arrow[r,"phi" swap] & |[alias=M22]| B \
endtikzcd
enddocument

As you can see more clearly in the second example, the advantage of this hybrid approach is that you have access to the full TikZ machinery while keeping the tikz-cd functionality.

TikZ is enough for (most) block figures.

documentclass[tikz,border=5mm]standalone
usepackageamsmath,amssymb
begindocument
begintikzpicture[>=latex]
path 
(0,0) node (M) $M_0,4$
+(180:2) node (B) $B$
++(90:1.5)+(0:.3) node (P) 
$M_0,4times mathbbP^1$;

draw[->] (B)--(M) node[below,midway]$kappa$; 
draw[<-,shorten >=2mm] (M.120)--(P-|M.120) 
node[left,midway]$pi$;
foreach i in 0,1,2
draw[->] (M.45)++(90:1mm)++(0:i*1.5mm)--+(90:.8);
draw[->] (M.45)++(90:1mm)++(0:3*1.5mm)--+(90:.8)
node[right,midway]$sigma_i$;
endtikzpicture
enddocument