Min datters tegning hænger på væggen i stuen.

Because min agrees with datter, not with tegning.

In Danish, the singular possessive forms are:

• min for common gender nouns
• mit for neuter nouns

Datter is a common gender noun, so you say min datter and therefore min datters tegning.

A useful comparison:

• min datters tegning = my daughter's drawing
• mit barns tegning = my child's drawing

Here barn is neuter, so it takes mit.

Datters is the genitive form of datter. Danish usually shows possession by adding -s directly to the noun:

• datter → datters
• Peter → Peters
• min ven → min vens

So min datters tegning literally means my daughter's drawing.

Unlike English, Danish normally does not use an apostrophe here. You just add -s.

After a possessor, Danish normally uses the noun in its indefinite form.

So you say:

• min datters tegning
• Peters bil
• hans bog

not:

• min datters tegningen
• Peters bilen
• hans bogen

This is similar to English: you say my daughter's drawing, not my daughter's the drawing.

So even if the meaning is specific, the noun form after a possessor is usually still the indefinite form.

That -en is the definite article attached to the end of the noun.

In Danish, instead of putting the before the noun like English does, Danish often adds the definite article as a suffix:

• væg = wall

• væggen = the wall

• stue = living room
• stuen = the living room

So:

• på væggen = on the wall
• i stuen = in the living room

It is the present tense of at hænge.

In this sentence, hænger describes the drawing's position or state, so it can correspond to English:

• hangs
• is hanging

Danish does not always make the same simple-present vs. present-continuous distinction that English does. So hænger can naturally cover both ideas, depending on context.

Because the two prepositions express different kinds of location.

• på is used for something on a surface: på væggen = on the wall
• i is used for something inside a space or room: i stuen = in the living room

So the sentence builds the location in two steps:

1. the drawing is on the wall
2. that wall is in the living room

Because Danish main clauses normally follow the V2 rule: the finite verb comes in the second position.

Here the first element is the whole subject phrase:

Min datters tegning

That entire phrase counts as one unit, so the verb comes next:

Min datters tegning hænger på væggen i stuen.

This is a very typical Danish sentence pattern.

Yes, but the verb still has to stay in second position in a main clause.

For example, you can front a location for emphasis:

• På væggen i stuen hænger min datters tegning.
• I stuen hænger min datters tegning på væggen.

These are all grammatical, but the original sentence is a very neutral, natural way to say it.

What changes is mainly the emphasis:

• starting with Min datters tegning focuses first on the drawing
• starting with På væggen i stuen focuses first on the location

Most naturally, i stuen narrows down væggen:

• på væggen i stuen = on the wall in the living room

So it sounds like the wall that is in the living room.

But in practice, it also tells you where the drawing is. A listener will understand the whole scene as being located in the living room. So the distinction is grammatical more than practical.

It often refers to a specific drawing that the speaker has in mind, but the context decides how specific it feels.

The important grammar point is this: after a possessor, Danish still uses the indefinite noun form:

• min datters tegning

That does not mean the reference is vague in the same way as plain en tegning. It just means Danish grammar requires the indefinite form after a possessor.

So the phrase can still be perfectly specific, just like English my daughter's drawing.