GAMs & homoscedasticity Despite my best Googling efforts, I can't find an answer to this simple question: 
Q: Does a generalised additive model with a link identity function and a Gaussian distribution assume homoscedasticity?
 A: Yes, the model that you mention assumes the model errors are independent, identically distributed with mean 0 and variance $\sigma^2$ (i.e., it assumes homoscedasticity).  If you fit your model with the gam function in the mgcv package, you can see that the model summary reports the estimated value of $\sigma^2$. For example:
library(mgcv)
set.seed(0)

n <- 20000
x <- (1:n)/n
set.seed(0)
e <- rnorm(n = length(x), mean = 0, sd = 0.15)
y <- 0.5 + 2*x + 3*x^2 + e


plot(x,y)

m <- gam(y ~ s(x), data=data.frame(x,y))

plot(m)

summary(m)

names(summary(m))

summary(m)$dispersion  # scale parameter reported by gam

#?summary.gam

0.15^2 # variance of errors used in simulation 

The estimated value of the standard deviation of the errors ($\sigma$) is usually referred to as the residual standard error. However, as explained by Gavin Simpson at How I can interpret GAM results?, the summary of the gam function reports the squared value of the residual standard error, which estimates the variance of the errors ($\sigma^2$).
