From what I understand the aggregates aren't self
Correct (per SN 22.59).
but now when a person dies
From the viewpoint of higher truth (that the aggregates are not self) there is no "person". The idea of a "person" is the same as the idea the aggregates are "self":
The Blessed One said, "And which is the burden? 'The five clung-to-aggregates,' it should be said. Which five? Form as a
clung-to-aggregate, feeling as a clung-to-aggregate, perception as a
clung-to-aggregate, fabrications as a clung-to-aggregate,
consciousness as a clung-to-aggregate. This, monks, is called the
burden.
And which is the carrier of the burden? 'The person,' it should be said. This venerable one with such a name, such a clan-name. This is
called the carrier of the burden.
SN 22.22
There are these five clung-to-aggregates, friend Visakha: form as a
clung-to-aggregate, feeling as a clung-to-aggregate, perception as a
clung-to-aggregate, fabrications as a clung-to-aggregate,
consciousness as a clung-to-aggregate. These five clung-to-aggregates
are the self-identification described by the Blessed One.
MN 44
Why now do you assume 'a being'? Mara, have you grasped a view? This is a heap of sheer constructions: Here no being is found.
Just as, with an assemblage of parts, The word 'chariot' is used, So, when the aggregates are present, There's the convention 'a being.'
SN 5.10
the material aggregates of his body dissolve what remains who gets reincarnated ?
This question is unrelated to higher Buddhism. In MN 117, it is taught there are two sorts of right view. The view of 'not-self' is transcendent right view and the view of 'self reaping results of kamma' is worldly right view. The Buddha did not mix up the two sorts of right view.
An analogy is kindergarten for children vs university for adults. Calculus is not taught to children in kindergarten and basic arithmetic is not taught to adults at university. Similarly, not-self is not taught to children and reincarnation is not taught to adults.
When the new material aggregates of the new body are born then they are different from the ones of the person who died .What defines that
THIS person is in THAT body?
In higher truth, the idea "a person" or "self" dies is called "annihilationism" and the idea "a person" or "self" is reincarnated is called "eternalism", as follows:
There are, bhikkhus, some recluses and brahmins who are eternalists, and who on four grounds proclaim the self and the world to be
eternal.
There are, bhikkhus, some recluses and brahmins who are annihilationists and who on seven grounds proclaim the annihilation,
destruction, and extermination of an existent being. And owing to
what, with reference to what, do these honorable recluses and brahmins
proclaim their views? Herein, bhikkhus, a certain recluse or a brahmin
asserts the following doctrine and view: 'The self, good sir, has
material form; it is composed of the four primary elements and
originates from father and mother. Since this self, good sir, is
annihilated and destroyed with the breakup of the body and does not
exist after death, at this point the self is completely
annihilated.' In this way some proclaim the annihilation, destruction,
and extermination of an existent being.
DN 1
In cultural Buddhism, because most faith follower Buddhists are not enlightened, they are taught and believe in "eternalism"; which is why they cling so tenaciously to metaphysical ideas of "rebirth".
In the history of Buddhism, there were heretic teachers, such as Buddhaghosa, who taught "not-self phenomena are reincarnated". This idea is heretical for at least two reasons:
To believe not-self is reincarnated destroys the incentive to do good personal kamma.
To believe not-self is reincarnated destroys the view of impermanence & dispassion required to realise Nibbana.
In the Pali suttas, it is clearly taught the same "person" inherits the results of "his" or "her" actions. That good & bad results follow from good & bad actions is what the scholars loosely translate as "rebirth" or "reappearance".