Learn Hardcore Turkish: Proje fikrini eskiz üzerinden kısaca anlattım. - I briefly explained the project idea through the sketch.

Proje fikri is a noun–noun compound meaning “project idea.” In such compounds the head noun (fikir) takes the 3rd person possessive marker -i, giving fikri.
Because this noun phrase is the specific direct object of the verb, it takes the accusative -(y)i. After a 3rd person possessive, Turkish uses the buffer consonant -n-, so: fikri + (ACC) → fikrini.
Rough breakdown: proje (modifier) + fikir (head) + -i (3SG poss/compound marker) + -n- (buffer) + -i (ACC).

Why not fikiri? Where did the vowel go?

Turkish has vowel loss (ünlü düşmesi) in some words when a vowel-initial suffix is added. fikir + -i → fikri (not “fikiri”). Other examples: şehir → şehri, burun → burnu.

Why not fikriyi for the accusative?

When a word already has the 3rd person possessive (-i), case endings attach with the buffer -n-, not -y-:

possessed + ACC: fikri → fikrini, kitabı → kitabını
vowel-final, not possessed + ACC uses -y-: kedi → kediyi

What exactly does eskiz üzerinden mean?

It’s a common postpositional phrase meaning “via/through/on the basis of a sketch,” i.e., you guided the explanation by referring to or going over the sketch. It is more than just physically “on top of”; it implies using the sketch as the organizing reference.

Should it be eskizin üzerinden? Why is there no suffix on eskiz?

With the postposition üzerinden meaning “via/through,” the complement is typically in bare form: X üzerinden.

eskiz üzerinden = “via a/the sketch” (idiomatic)
eskizin üzerinden is more literal/physical: “from the top of the sketch” (e.g., you lifted something off it).
With pronouns, you must use genitive + possessed form: benim üzerimden, onun üzerinden.

Could I use üzerinde instead of üzerinden?

üzerinden = “via/through/by way of” (procedural/mediated sense).
üzerinde = “on (top of)/on” and in some idioms “on/with respect to” (e.g., konu üzerinde konuşmak = “to discuss a topic”).
In your sentence, üzerinden is the natural choice for “by using the sketch as the basis.” Üzerinde would suggest physically “on the sketch,” or a different idiomatic frame.

What’s the difference between eskiz üzerinden and eskizle?

eskiz üzerinden highlights “going over/through the sketch,” using it as the organizational reference.
eskizle (with -le, “with/by”) frames the sketch more as a tool/means. Both can be fine; üzerinden is slightly more “by going over it,” whereas -le is more neutral instrumentality.

Is the word order fixed? Where can kısaca go?

Turkish is flexible. Common placements:

Proje fikrini eskiz üzerinden kısaca anlattım.
Proje fikrini kısaca eskiz üzerinden anlattım.
Kısaca, proje fikrini eskiz üzerinden anlattım. Placing kısaca earlier foregrounds the brevity; keeping it near the verb is also natural.

What does kısaca modify, and are there alternatives?

kısaca is an adverb modifying the manner of explaining: “briefly.” Alternatives include kısaca olarak (less idiomatic), kısa bir şekilde (wordier), özetle (“in summary,” slightly different nuance). Kısaca is the most natural here.

What tense/person is anlattım?

Stem: anlat- (to explain)
Past definite: -dı → voicing to -tı after t
1st person singular: -m
So anlattım = “I explained.”
Related contrasts:
anlatıyordum = “I was explaining”
anlatmıştım = “I had explained”
anlatmışım = “apparently I’ve explained” (reportative)

How does definiteness work here without “the”? Why is the object marked?

Turkish marks a specific/definite direct object with the accusative. Proje fikrini is definite/specific, so it’s marked. If you meant “a project idea” (indefinite), you would use no accusative: Bir proje fikri anlattım (“I explained a project idea”).

Can I pluralize or make it indefinite? For example, “through sketches” or “through a sketch.”

Plural: eskizler üzerinden = “via sketches” (fine if you actually used multiple sketches).
Indefinite singular: add bir if you want to stress “a” sketch: bir eskiz üzerinden. In many contexts, bare eskiz üzerinden already works as “via a/the sketch” depending on context.

Is there any difference between üzerinden and üstünden?

Both can mean “over/from/through,” but:

üzerinden is more neutral/formal and common in abstract uses like “via/through (a medium).”
üstünden is more colloquial and often feels more physical or idiomatic (e.g., üstünden geçmek = “go over/review, pass over”). In your sentence, üzerinden is preferred.