Can the methods of classical algebraic geometry be made rigorous with a synthetic approach? There are approaches to real analysis that use an axiomatization of nilpotent infinitesimals to enable rigorous synthetic reasoning about infinitesimals, which is arguably closer to the reasoning employed by mathematicians prior to the arithmetization of analysis.
While I am far from an expert, it seems that the conventional narrative is that increasing doubts about the validity of some of the results of classical algebraic geometry led to a similar arithmetization of algebraic geometry by Zariski and Weil. The gap between the resulting methods and the underlying geometric reasoning seems a bit wider than the gap between $\epsilon$-$\delta$ arguments and infinitesimals, although that is entirely personal opinion and may only be due to my familiarity with the latter.
Is it possible to do algebraic geometry in a synthetic manner that enables rigorous reasoning but is closer to the style of argument employed by classical algebraic geometers?
 A: I am not sure what you would call synthetic algebraic geometry. Nevertheless, using some methods which are somtimes called synthetic, Fyodor Zak magnificently reproves and improves many results claimed by italian geometers. In particular his classification of Severi varieties (see : http://mathecon.cemi.rssi.ru/zak/files/Zak_TSAV.pdf) could certainly be considered as a masterpiece of modern synthetic projective geometry.
A: 
Is it possible to do algebraic geometry in a synthetic manner that enables rigorous reasoning but is closer to the style of argument employed by classical algebraic geometers?

I sure hope so. You can have a look at notes of mine which develop the basics of a synthetic account of algebraic geometry. Especially relevant is Section 20, which presents a couple of case studies, including computing the cohomology of Serre's twisting sheaves.
To give a short teaser: Synthetically, we can define the projective space $\mathbb{P}(V)$ associated to a vector space $V$ simply as the set of one-dimensional subspaces of $V$. The twisting "sheaf" $\mathcal{O}(-1)$ is then simply the family $(\ell)_{\ell \in \mathbb{P}(V)}$ of vector spaces. Its dual sheaf is the family $(\ell^\vee)_{\ell \in \mathbb{P}(V)}$. The scheme structure is automatically taken caren of.
However there is still much more to be done. The most pressing concerns are maybe:


*

*There should be an easy and intuitive synthetic description of proper morphisms. Right now we do have a synthetic description, but it's very close the usual non-synthetic description and doesn't exploit the unique possibilites of the synthetic context.

*We have to develop a synthetic account of cohomology, derived categories and intersection theory.

